I'm confused!
If your daughter's job will still exist when the business changes hand she IS entitled to protection under the TUPE rules but she's NOT entitled to a redundancy payment.
If your daughter's job will genuinely cease to exist when the business changes hands then she's NOT covered by the TUPE rules but she IS entitled to a redundancy payment.
I can't see any way that your daughter could seek a redundancy payment after three months of working with the new employer (after her rights had earlier been transferred under TUPE) unless, by a strange coincidence, the new employer just happened to reorganise the company (with the subsequent loss of her role) at that time.