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The illusion has the appearance of the caster when they cast mislead

The description of the Mislead spell does not contain any reference to the original or true aspect of the caster, nor does it contain any particular rule for interacting with other spells, in particular those from the Illusion school.

Moreover, in the PHB, page 205, one can find the rules for combining magical effects:

The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. [...]

We have thus two different spells, which are doing two different things: the first sentence assures us that both their effects apply during their overlapping duration.

Mislead specifies that an illusory double appears in the same space where the caster is standing (emphasis mine):

You become invisible at the same time that an illusory double of you appears where you are standing.

The intent and the meaning of this is to provide no evidence¹ that the apparent creature that is moving and acting like the caster is actually an illusory double. If the modifications provided by Disguise Self were bypassed by Mislead, then the latter spell would have little use in crowded places (like scouting the royal palace during the Gran Gala, or running away from the deadly fight with Rocs). Moreover, Mislead is a 5th level spell: it would be not so useful if it couldn't interact well with lower-level spells such as Disguise Self.

¹ Note that there is only the Somatic component for this spell.

The illusion has the appearance of the caster when they cast mislead

The description of the Mislead spell does not contain any reference to the original or true aspect of the caster, nor does it contain any particular rule for interacting with other spells, in particular those from the Illusion school.

Moreover, in the PHB, page 205, one can find the rules for combining magical effects:

The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. [...]

We have thus two different spells, which are doing two different things: the first sentence assures us that both their effects apply during their overlapping duration.

Mislead specifies that an illusory double appears in the same space where the caster is standing (emphasis mine):

You become invisible at the same time that an illusory double of you appears where you are standing.

The intent and the meaning of this is to provide no evidence¹ that the apparent creature that is moving and acting like the caster is actually an illusory double. If the modifications provided by Disguise Self were bypassed by Mislead, then the latter spell would have little use in crowded places (like scouting the royal palace during the Gran Gala, or running away from the deadly fight with Rocs). Moreover, Mislead is a 5th level spell: it would be not so useful if it couldn't interact well with lower-level spells such as Disguise Self.

Can one change the appearance of the double?

The spell description does not provide such option: for comparison, see the Alter Self spell (emphasis mine):

Change Appearance. You transform your appearance. You decide what you look like, including your height, weight, facial features, sound of your voice, hair length, coloration, and distinguishing characteristics, if any. [...] At any time for the duration of the spell, you can use your action to change your appearance in this way again.

In this case, the option to modify the appearance when the spell takes place is clearly stated.

¹ Note that there is only the Somatic component for this spell.

The illusion has the appearance of the caster when they cast mislead

The description of the mislead spell does not contain any reference to the original or true aspect of the caster, nor does it contain any particular rule for interacting with other spells, in particular those from the Illusion school.

Moreover, in the PHB, page 205, one can find the rules for combining magical effects:

The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. [...]

We have thus two different spells, which are doing two different things: the first sentence assures us that both their effects apply during their overlapping duration.

Mislead specifies that an illusory double appears in the same space where the caster is standing (emphasis mine):

You become invisible at the same time that an illusory double of you appears where you are standing.

The intent and the meaning of this is to provide no evidence¹ that the apparent creature that is moving and acting like the caster is actually an illusory double. If the modifications provided by disguise self were bypassed by mislead, then the latter spell would have little use in crowded places (like scouting the royal palace during the Gran Gala, or running away from the deadly fight with Rocs). Moreover, mislead is a 5th level spell: it would be not so useful if it couldn't interact well with lower-level spells such as disguise self.

¹ _{Note that there is only the Somatic component for this spell.}

The illusion has the appearance of the caster when they cast mislead

The description of the Mislead spell does not contain any reference to the original or true aspect of the caster, nor does it contain any particular rule for interacting with other spells, in particular those from the Illusion school.

Moreover, in the PHB, page 205, one can find the rules for combining magical effects:

The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. [...]

We have thus two different spells, which are doing two different things: the first sentence assures us that both their effects apply during their overlapping duration.

Mislead specifies that an illusory double appears in the same space where the caster is standing (emphasis mine):

You become invisible at the same time that an illusory double of you appears where you are standing.

The intent and the meaning of this is to provide no evidence¹ that the apparent creature that is moving and acting like the caster is actually an illusory double. If the modifications provided by Disguise Self were bypassed by Mislead, then the latter spell would have little use in crowded places (like scouting the royal palace during the Gran Gala, or running away from the deadly fight with Rocs). Moreover, Mislead is a 5th level spell: it would be not so useful if it couldn't interact well with lower-level spells such as Disguise Self.

¹ Note that there is only the Somatic component for this spell.

The illusion has the appearance of the caster when they cast Mislead

The description of Mislead spell does not contain any reference to the original or true aspect of the caster, neither it contains any particular rule for interacting with other spells, in particular from the Illusion school.

Moreover, in the PHB, page 205, one can find the rules for combining magical effects:

The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. [...]

We have hence two different spells, which are doing two different things: the first sentences assures us that their effects apply during their duration.

It specifies that an illusory double appears in the same space where the caster is standing (emphasis mine):

You become invisible at the same time that an illusory double of you appears where you are standing.

The intent and the meaning of this is to do not provide any evidence¹ that the creature that is moving and acting as the caster is actually an illusory double. If the modifications provided by Disguise Self were bypassed by Mislead, then the latter has little use in crowded places (like scouting the royal palace during the Gran Gala, or running away from the deadly fight with Rocs). Moreover, Mislead is a 5th level spell: it could be not so useful if it couldn't interact with lower level spells such as Disguise Self.

¹ Note that there is only the Somatic component for this spell.

The illusion has the appearance of the caster when they cast mislead

The description of the mislead spell does not contain any reference to the original or true aspect of the caster, nor does it contain any particular rule for interacting with other spells, in particular those from the Illusion school.

Moreover, in the PHB, page 205, one can find the rules for combining magical effects:

The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. [...]

We have thus two different spells, which are doing two different things: the first sentence assures us that both their effects apply during their overlapping duration.

Mislead specifies that an illusory double appears in the same space where the caster is standing (emphasis mine):

You become invisible at the same time that an illusory double of you appears where you are standing.

The intent and the meaning of this is to provide no evidence¹ that the apparent creature that is moving and acting like the caster is actually an illusory double. If the modifications provided by disguise self were bypassed by mislead, then the latter spell would have little use in crowded places (like scouting the royal palace during the Gran Gala, or running away from the deadly fight with Rocs). Moreover, mislead is a 5th level spell: it would be not so useful if it couldn't interact well with lower-level spells such as disguise self.

¹ _{Note that there is only the Somatic component for this spell.}