Considering how unwilling he has been to motivate himself to learn how to look after his child, and cannot do it without constant instructions, encouragement and leadership from you - I seriously doubt he will motivate himself to seek contact in a court. And you mentioned yourself your fear that if you leave, he will just not bother to seek any further contact with your child.
I think you really do have his number, sadly and hard though it is to deal with.
This is where it may help you to share with your HV, when ready and on your terms, your concerns and what has happened in the past few days, and how this has driven your decision to move out in part because you have concerns for your child's safety. Texts would be helpful evidence to keep of this, plus a diary of what happened and when. If he is sufficiently motivated - and again, it's telling to me that you see the money as being what may drive him to do this, and not just wanting to continue parenting his child - to go to court, then you will have the evidence that there were major safeguarding concerns. Even if you leave the UK, you may still be required to organise contact, and this evidence would open doors for you to do things like require contact to start under supervision in a contact centre, where he was observed in how he cared for dc, how he interacted and met needs. The observers would need to sign off that he was consistently doing a good enough job in this case before moving to letting him have unsupervised contact, which would be a weight off your mind.
I can only imagine how hard this is right now, but please do think of organising an evidence trail now. It's easier to have it and not need it than to later wish you had.