I do not see how, someone paying half the day to day running expenses of a house they live in should have any legal claim on the house itself. If they are contributing to the mortgage and/or make major improvements at their expense that is different.
SJ23, I suggest you talk to the CAB or Age UK and get clear advice whether getting your partner to pay his share of the bills affects your full ownership of the property or not. If not, open a joint household bank account, calculate what the bills ar, divide by 2 and you then both put that amount of money in the bank each month. He can pay utility bills without the bills being in his name. I am the family finance manager and pay all the bills, but some are in my name and some in DH's.
I think having lived at home until his parents died and then moved in with you he has expected to live with you as he lived with his parents. Personally, I would tell him that two years is long enough to live with you temporally and now is the time for him to buy a property and stand on his own two feet. If he kicks up a fuss or breaaks it off, you will know that he was with you merely for the domestic comforts of a settled home like that he had known with his parents.