This is not an easy one and in replying, I must bear in mind that this is not an examination question - it''s a practical issue. There may be things you could do, but you are, or could be, in the realms of an area of law renowned for its complexity - and if you had the money to fight, then maybe that would be better spent housing yourself then lining lawyers'' pockets.
In a word, if you are not an owner of the house, and have no interest in it, then you have no right to live there and can be ordered to leave on giving reasonable notice ; and perhaps, in your case, given that you have lived there a long time and have a child, that notice could be longish, a month at minimum, perhaps more.
The Courts have, over time, evolved a series of principles called proprietary estoppel, constructive trusts and resulting trusts. This sounds a bit of a mouthful and believe me, it is, but to over simplify, it is possible to establish an interest in a house by means like paying the mortgage, financial improvements, or do-it-yourselfing. Now your first reaction might be to say, " Yes !! That''s me ! " but your partner may not see it the same way and the issue may take litigation to resolve. Could you afford that ?
Another possible route, and perhaps a simpler one, is Section 15 and Schedule 1 to the Children Act 1989. Under this legislation it is possible for a Court to order the transfer of property for the benefit of a child, and it could, in theory, be used to give you and your child the right to stay in the home while he is under age. But it seems that this section is not all that much used, and quite honestly you''d need advice from a lawyer who specialises in such matters. And again there is the issue of expense.
A third option, and a sphere of law with which I am much more familiar, and there was a time, I think, when I could fairly be described as an expert - is to avail yourself of the homeless provisions in the Housing Acts. In other words, there is a very strong possibility that you could be entitled to rehousing by the local authority. If rent is the issue, then housing benefit would be available to assist.
I must say that I think your partner must at least face up to his responsibilities as a parent. You may well be entitled to local authority re-housing, but it''s a bit hit-and-miss what you end up with ; in the short term it can be a hostel ; some are good, some not so good, and it''s better to look for accommodation on your own account, but given that you have a child and may be on benefits, this may be easier said than done and I used to advise people that local authority re-housing is really the last option to examine. But it''s better than the alternative.
I think that the Children Act might possibly be your best bet, but there is the issue of expense. I don''t know your partner, but morally speaking it would be reprehensible in the extreme to put his child on the street. I think that, at this stage, my advice would be to try and get half an hour''s free advice from a family lawyer and made it clear that you want advice about the possibility of a transfer under the Children Act.