mikado my heart goes out to you.
The stuff below, I hope some of this helps ... not sure how much will resonate with you or if it will feel like it doesn't apply.
I'm afraid that the unbearable truth is that unless your mother is honest with you, it's most unlikely that it'll ever be possible to find your father. It's a brick wall that you cannot get through. I hope that one day she will have the grace to give you the info that she does know. If that happens you have the Salvation Army, also when I was tracing my birth-parents Social Services helped greatly though that was quite some time ago now and they may not do that any more, and lastly there are (reputable) private investigators. And the internet ofc.
But it does have to be faced that she may not ever give you more information. As you clearly realise, she may not have more than the tiniest bit :( Also .... if you did ever find him, he might be someone you'd wish you'd never contacted. Finding a lost parent can be good, but it can be bloody awful too and facing the reality can be extremely unwelcome (not that I'm speaking from experience coughs).
It's clear that you feel such yearning, not only for some information on him but for real, unconditional love from your mother.
Again, I'm very sorry to say that it seems possible that you will never get that from her. I suspect that you know that at some level.
How to cope? it's always difficult. When you are left alone in the world without family and with a yearning for them, a circle of friends -strong, real friends who are there through thick and thin- do help. It's not the same, but it's something.
Other than that, some intelligent and skilled therapy might help, possibly group therapy? there's therapists and therapist though. One registered with the main psychotherapy organisations should have some chance of being good.
If you can build your own family, partner and children, that will be a massive thing. It is building your own family who can't replace the one you came from but they become profoundly central to your life. As long as the partner is decent and treats you well! But, well, not everyone does find a good life-partner so it isn't something I could say you could rely on.
Those first two ideas would help a little, I hope. Beyond that, sometimes there is no actual resolution to an unsatisfied deep need like that for family. But what can happen is that over time, and it does take time, years, you grow stronger. You learn to live with the longing and unsatisfied curiosity and while it never goes away, you build up other things that are just as strong a part of you. They are real and solid and good. Career, family of your own, good friends.
I think you have to face that you may not ever know and that your mother is not loving towards you. You may have to live with that longing for family, but choose to turn your energies into other things and build them up instead. Find something that really matters to you and that you can contribute to, so that when you look back you know you have made a difference to someone or some people. Its not the same, but it does actually help.
Either way, I really hope you can come to terms with it all, whatever happens with your mum.