Not for environmental health noise complaints, no.
But OP does if she want help with her tenancy/neighbour does if OP wants her evicted for breaching her tenancy contract by causing harassment and antisocial behaviour.
OP contact your LL (assuming you and neighbor share the same LL) and ask what evidence they need, then set out to provide it. It's possibly going to be 2yrs of evidence because evictions from housing association/council property is a very last resort.
Get help from your GP for your MH. If you can't change the situation you need a way of coping with it.
You say family are unreliable. Family, his and yours, are who will first be approached to take on the DC if you turn them over into care. Being in a children's home or with a string of foster carers isn't necessarily going to be any less damaging for them than living in this situation with you. Ditto being deprived of you if they're placed with his family who decides to make things difficult for you.
If you make yourself homeless what do you plan to do for yourself? Because the council won't house you, you'll be left to secure private rental. I appreciate this could be easier as a single effectively childfree person who could rent a room, but will you really be any happier, separated from your DC?
Can you afford to private rent anything at all? I'd move with DC into a one bedroom flat if necessary/possible. I suspect life is unlikely to improve there if multiple neighbors are bullying you. Unless you feel you could ignore the rest if this one was dealt with.