I responded page 1 of this but i see things have moved on a bit with the car.
OP - it's her car so her responsibility to get it booked in and taken for MOT.
Leave that to her. Check online (you only need reg number to check the official gov site and it's free) to see if she has done it.
You can also see the pass/fail and repairs needed.
If it lapses, you will see that.
You could choose to report her if you believe she is still driving in it and you want to save her from the catastrophe of causing an accident uninsured and no MOT.
Or you can let the Police catch her. (Although it's lapsed insurance that triggers on ANPR cameras; I don't think no MOT does even though her insurance will be nul and void)
You really do need to let them go and face the consequences as adults. We did the financial help thing a bit differently. We set each of ours off with about £10k and that was for all years.
So - could that be an option? Work out the support you're willing to pay over a year and release as a lump sum for her to budget? Give her some responsibility.
My older 2 managed ok (even the 28yr old referred to on page 1 here)
But DS 3 blew his 10k in year 1. We did not bail him out. He panicked and got work. Now he has left Uni and is so much more cautious with money. He even saves!
Sometimes the hardest lessons are what's most needed.