Rights when separated and buying out house

Hi. I am actually writing this on behalf of my mentally ill sister.
Her circumstances are this:-
4.5 years ago she was sectioned, leaving her long term partner and their 4 children. For 1.5 years the hospital and her tried to work to get her to a position where she could return to family life, but alas it could never happen. So 3 years ago she declared to her then partner it was over.
Facts:-
Not married
Own house as joint tenants
He has continued to live there with children
She put more in up front
She has never worked during their time together (due to illness) bur was a stay at home mum
He has continued to pay mortgage with no financial contribution from my sister in 3 years since split.
They have a repayment mortgage for the most of it, however a small element was interest only as they’d planned for an endowment he had to pay this bit off.
Only now 3 children left at home (16 and 2x14). 17yo left home.

They had come to a verbal agreement of how much she would accept to be bought out as he now wants to do so.
However sadly he has had a change of heart and she received a letter this week via a solicitor offering substantially less. He has undervalued house (with no proof of valuations), said they were separated a year longer than they were, trying to claim for costs of improvements since she left (new carpets etc) despite not asking her at time.
So my questions are:-
1. Are children taken into account when working out a split for buying her out ( I know they have a right to roof over their heads, so if he can’t afford it then she will wait until kids 18)
2. Can he claim for the 3 years of mortgage payments since they split up?
3. Can he ask her to contribute to costs she wasn’t consulted about for home improvements
4. Does it matter if she put more in to begin with, does this count for anything?
5. Assume it’s tough luck about the plans for using his endowment to pay interest only section?

I will also be asking solicitor for 3 valuations from past month, asking when she can do her own valuations,
Also possibly receipts for all improvements.

I don’t want her (read me on her behalf) to have go to war on this, but I obviously won’t let be taken advantage of.
My idea scenario is to chuck enough facts at him to show him I know what I’m taking about (I dealt with my own divorce financials so not completely clueless, but this is different) and make him realise the original deal was better than the one he gets if it turns nasty and she’s no longer willing to accept what she said she would (I believe she’d be entitled to more than original verbal agreement).
Thanks all in advance

Ah, not married and committed, as much as it sounds harsh, keep emotions out of the calculation.

You cannot pay back the time without children and in the care system.

The two legal systems are Family Law and ToLata.en.m.wikipedia.org/wiki/Trusts_of_Land_a...of_Trustees_Act_1996

Homeless?
Under the new Housing act, the council should now house her as a vulnerable person. She just needs to present her self as homeless, ask a social worker to help or shelter.

Share of home
I think even if not married, for the benefit of the children, a court will give her nothing now. Zero, until the kids are old enough to leave the home. A fair judge/will continue current arrangement. If he can pay she will get the money.

Legal right to look, atfer children
It would have to be a very pro-woman judge to see that she got anything until at least the children have left home. A judge looking at the law may see a man with no parental responsibility (PR) looking after children and say please leave. Until he applies for PR.

Questions
So it's practically a yes to all your questions. And that's how you should do this. Place all the calculations in a table, and see how the money looks, for a buyout. Start with valuation then divide by share percentage. See how much he has put into property, take this off her payout. then see if this is enough to buy somewhere. Also if he has no money to buy her out in a lump sum, I'm not even sure he will be able to pay her off bit by bit.
However, she will be able to get her money back in two ways I imagine.
1. sever the tenancy and sell her side to someone.
2. sever the tenancy and wait. She will eventually be paid out.

Prohibitive steps to evict the father.
On the other hand, she could ask the court to evict him, if she is the only person who has PR as an unmarried father usually does not. She legally has the only claim over the children and therefore the home. she then takes care of the children, and he has to find somewhere to live.

the only real leverage is "hassle", try to explain what you want and how much it is a hassle to go through the court, not all the above, but just how much you want. How much court costs and then draw up an agreement.

Then go to court and ask for the money from the home to be repaid.

I'm not a lawyer: Land law is very complex. Family law is easier. Make an offer, and try to negotiate. Then take that agreement or disagreement to court. You have already created a trust to buy a property in parts, it will be important to see how severing that trust will help you at court, or in the future.

In any event, you should register your trust with the land registry, so that if the house is sold, you will get your share.

Good luck.