OK, I'm not a lawyer, but I have read up on this because I used to work with teenagers. But since nobody else has replied, this is my understanding as to what the law says.
Assuming both you and your ex have parental responsibility for your son, she is entitled to make decisions for and about him, without consulting you. And you can make decisions without consulting her. There are exceptions - I think if one parent wanted to take the child out of the country and there was a risk they might not bring them back, the other parent can apply to the courts to forbid this - but generally anyone with parental responsibility can act independently of the other(s).
But I doubt parental permission is needed anyway, if the GF's mum is OK with it. I'm not a police officer either, but I think the police have powers to bring a child/young person (under 16) home if requested by a parent. So, if you didn't want him to stay at the GF's, in principle you could ask the police to bring him back. But what happens next? He hasn't broken any laws, so neither you nor the police have any right to restrain him. He could just go straight back round to her place and I imagine the police might get fed up with repeated requests. I suspect what they would do is make sure he's safe and then ask him what he wants to do. If he doesn't want to go home then I think they would leave him - which is what I understand they would do automatically once he turns sixteen.
As for sharing a bed with the GF, I'm pretty sure that isn't illegal in itself. Possibly there may be a law about siblings sharing a bed - I really don't know. But not BF/GF. And I think more and more parents are becoming comfortable with it, particularly in the Netherlands (I read that somewhere).
Of course, if they do anything sexual (which they might not - I think lots of teens do just want to literally sleep together) then technically it is illegal. But "anything sexual" includes kissing, which they've probably done anyway. They won't get into trouble. It was made clear when the 2003 act was drafted that the law was intended to protect young people, not to punish them. It would not be in the public interest to do so.
Personally, I think the law could have been written in a different way - as many other countries do - such that two 15-year-olds kissing are not automatically classed as criminals, but it wasn't.
If your concern is that you, your ex, or the GF's mum could be accused of aiding and abetting under-age sex, I think it is unlikely. The legislation following the Gillick case (1985?) which protected healthcare professionals from prosecution for providing contraception to under-16s also applies to anyone involved with children/young people. Social workers, teachers, parents. The aim is always one of safety. The GF's mum could justifiably claim that she didn't want her daughter to have sex under a bush in the park, and that home in bed is a safer option. Difficult to argue with - except irrationally.
The only problem for your son and his girlfriend is if they are still together when one of them turns 16 (and the other doesn't). While they are both under 16 they are both technically breaking the law (if they are sexually active). Once one turns 16 that one becomes the "perpetrator" (and is a criminal) while the other is the "victim" (and is not a criminal). Then, when the younger one turns 16 it all becomes completely legal. All this in the same relationship. This is one of the reasons I think the law could have been better written.
This is a good website for your son and his girlfriend to look at.
www.themix.org.uk
Hope this helps.