OK OP the situation you are describing is a bit like this:
House worth 120k. Equity = 76k (minus the 19k council and 25k mortgage). Currently all in his name (loan and equity). From a protection point of view, he can write a will leaving you the house, and take out mortgage protection insurance if he doesn't already have it, but he can obviously change that at any point.
So to clear the mortgage and charge you would need to get a new mortgage of 44k. How much could you borrow on your income? How much can you borrow jointly? How much of a cash deposit have you saved?
You say he's older so you're probably looking at a shorter term, say 15 years. Assuming you can jointly borrow 44k over 15 years, that's what I would do, minus any cash contribution you have saved, then have your name added to the deeds as joint tenants, and pay all mortgage repayments jointly going forward.
Then the question is, how do you pay him back 'his' share, and how do you protect yourself? You say you have already been living there and contributing. So: does he think any of the equity in the house is yours? Do you have your own cash deposit? If you want to buy out 'his' share the sensible thing to do is to use any cash you have towards that. So if you are going to be a 50% owner, 'his' share is worth 38k. But if you've contributed, presumably he'd be willing to reduce that, the question is to what - 30k? 25K?
So I guess there are different ways to do it. If you have cash, and can secure a joint mortgage, I would set up a transaction where you use your cash to pay off the mortgage then jointly remortgage for a lower amount, and what you contribute in cash comes off the 38k (or whatever amount you agree) you 'owe' him for a 50/50 stake. Then you could simply agree that you will pay all the mortgage repayments for the first year or two (depending on what they work out at) until you reach the agreed amount, then you will be even, then you can start splitting it again.
That is one way, it sounds a bit complicated but so is the situation you're describing. I think for security, you want to jointly own the house: you could own it in shares (as tenants in common) but then he can leave his half where he wants if he dies. Think about what is acceptable to you both, and particularly think about how much he wants for his 'share' and how much you can realistically finance. Then get proper advise.