"T" messed up and processed a distribution, not a trustee-to-trustee transfer despite what appears to have been a proper request to do a trustee-to-trustee transfer and having prepared the check appropriately (although it would have been more proper to have the check made out specifically to the receiving IRA).
If this had been done as a trustee-to-trustee transfer of a traditional IRA to another traditional IRA, no Form 1099-R should have been issued and no taxes should have been withheld. (Code G is never to be used for the movement of funds between IRAs.)
Because taxes were withheld, it would be difficult to treat get the reporting corrected to show that only the 10% withheld for taxes was the gross amount distributed. Of course this makes your wife subject to the one-rollover per 12-months limitation, so hopefully she rolled over no other IRA distributions in the one-year period ending on the date of this distribution and will not do so in the one year period beginning on the date of this distribution.
I would approach "E" (or any other IRA custodian, even back to "T" if they are willing to help correct theire mistake) to see if they will accept your wife's certification under IRS Rev. Proc. 2016-47 as modified by Rev. Proc. 2020-46 that rolling over the 10% withheld for taxes would qualify for a waiver of the 60-day rollover deadline due to financial-institution error and allow her to complete the rollover of the entire gross amount. That way the early-distribution penalty would be avoided and taxes would continue to be deferred on the amount that was withheld. Rolling over an amount equal to the tax withholding would still be a rollover of a single distribution and would not violate the one-rollover-per-12-months limitation.
Whatever the result with the Form 1099-R, you'll need to report whatever portion of gross amount that ultimately made it to the new IRA as having been rolled over.
With potentially serious mistakes like this, perhaps it is good that your wife has moved the IRA from "T".
