As far as I'm aware, you can't realise or release equity in a mortgaged property.
If you've had periods of not being able to work because you've been the sahp and are only able to work 20hrs per week because you're the primary carer and will need to continue providing a home for the dc, I would suggest you endeavour to negotiate a split of the current equity along the lines of what would be reasonable if you were married.
Approximately how much equity could be realised if the property was sold now and are you confident that the mortgagee(s) would consent to you becoming the sole mortgage holder if his name is removed, or would you need to remortgage/seek another mortagee? Did you pay a substantial amount of the deposit (if any) and would you be able to buy him out now if the split was 70/30?
If he continues to pay his share of the mortage repayments he will of course be entitled to his share of the equity when the house is sold but if, for example, you are proposing to pay all of the mortgage repayments for the next 5 or more years it will be necessary peg his share of the equity (with a rate of interest to be agreed) to the sum he could expect to realise on the date you take over the payments.
If his solicitor is offering 5 years and you are confident that you could get ahead in that time and buy him out as per the above paragraph, this may be the best way forward for all concerned but, needless to say, any such agreement should be watertight and fire/bombproof and should take into account unforeseen/unlikely eventualities such as the death of one or both of the parties to it. To my mind it would be preferable to have a clean break, as it were, now.
There's a lot to be considered and it's not possible to give detailed advice without knowing all of the facts and figures. However, I would suggest that you talk to your solicitor about the possibility of applying on behalf of the dc for a share of the property under the Children Act 1989 as, if nothing else, this could be used by way of a bargaining chip.
What it with these men? From what you've said, if your name wasn't on the mortgage I reckon he'd have you out on the street or into some inferior rented accomodation without a second thought.