Would someone who is casting something with a Somatic and Material component, say Minor Illusion, also be able to cast something with only a Verbal component, say Thaumaturgy, in the same action? I'm making a character who will be casting these cantrips on a regular basis for cosmetic changes, and want to know if there is an official ruling on this.
No, the whole action is needed to cast a spell with a casting time of 1 action
There is nothing in the rules for casting spells that suggests that you can cast two spells at once with one action.
If you use your action to cast a spell with a casting time of 1 action, you cannot cast another spell with a casting time of one action unless you have another action (such as via the fighter's Action Surge), or unless you can modify the casting time (such as via the sorcerer's Quickened Spell), regardless of each spell's components.
“Cosmetic changes” seems like it will usually be an outside-of-combat concern, in which case your concern shouldn’t be much of an issue.
The action economy of spell casting times is mostly a concern in combat. In combat, your action can be used to cast at most one spell with a casting time of 1 action.
Outside of combat, the rigid structure of initiative and action economies usually matters little, so you could easily cast multiple spells in succession with little issue, as long as concentration rules are observed:
You lose concentration on a spell if you cast another spell that requires concentration. You can't concentrate on two spells at once.
Now if you really want to cast thaumaturgy and minor illusion in a single turn, the sorcerer metamagic quickened spell has what you need:
When you cast a spell that has a casting time of 1 action, you can spend 2 sorcery points to change the casting time to 1 bonus action for this casting.
The Divine Soul Sorcerer allows you to access thaumaturgy as a sorcerer, so this is a feasible option for what you wish to achieve.