Cape of the Mountebank (DMG 157)
While wearing it, you can use it to cast the dimension door spell as an action.
Activating an Item (DMG 141)
Spells Some magic items allow the user to cast a spell from the item.
The spell is cast at the lowest possible spell and caster level,
doesn't expend any of the user's spell slots, and requires no
components, unless the item's description says otherwise. The spell
uses its normal casting time, range, and duration, and the user of the
item must concentrate if the spell requires concentration.
Ready (PHB 193)
Sometimes you want to get the jump on a foe or wait for a particular circumstance
before you act. To do so, you can take the Ready action on your turn,
which lets you act using your reaction before the start of your next
turn. When you ready a spell, you cast it as normal but hold its
energy, which you release with your reactions when the trigger
occurs. To be readied, a spell must have a casting time of 1 action,
and holding onto the spell's magic requires concentration.
Using the above quotes (with some unrelated text omitted, and with my emphasis in bold), I see this situation with the following logic:
- The cape says it allows the user to cast a spell.
- The rules for activating a magic item explicitly refer to casting a spell from the item.
- Those same rules say that most the normal rules for casting apply, including concentration.
- Readying the spell requires casting the spell, and holding it with your concentration.
- You have until the start of your next turn to use a readied item. Otherwise, you lose the spell.
Therefore, if the trigger does not occur, the spell is lost. You must wait until the next dawn to use it again.
Note that this question explicitly asks what happens when the effect of a magic item is held via the Ready action. For the more general case of readying the use of a magic item, see this question.