Who gets to buy who out?

[TheLasrStraw]

Mum who can raise cash or dad who would need a mortgage?

Assume child will be over 18 and at uni.

I don't think either of those two factors decide. The two adults need to agree, possibly based on what's best for the child, or they sell up if they can't.

Doesn’t matter if it’s cash or mortgage. If both can afford to take on the house and raise the funds to buy out the others share.
A house is just a house though - a home is where you make it

You either decide together who is buying out the other or a judge makes the decision for you. This would be part of the asset pool division if you're married. If you go to court the judge may decide house should be sold and equity split.

Yes I was also about to say the same as PP. either the adults agree between them or the judge decides.