Yes job seekers is different to UC in terms of eligibility.
I'm sorry you're dealing with this
Apart from everything else, savings are meant to be there for actual emergencies. Having none left between you is extremely risky, and he is being absolutely reckless with all your lives. You have a child now for goodness sake.
Honestly, why not go and see a divorce lawyer and see what they advise you? Gather more information so you can at least make an informed choice.
My unprofessional suggestion:
You said he is emotionally invested in the house, so what if you call his bluff and make absolutely 100% clear that if he doesn't get a job - any job - soon, you would rather default on the mortgage than continue this way. He might get his arse into gear. Or not. But if he's really adamant to not get a job, you'll find yourself in the same situation of defaulting when your money runs out, just less your savings. Which you will need.
So while this strategy might seem risky, it's not much riskier than just waiting to see how things pan out if you cough up your savings. Maybe seeing a red letter warning will scare him - so why not do this sooner rather than later? Most people will only push you as far as they can get away with.
I'd also start booking Estate Agents to come and list the place. Book one to come when you're both in, book another when it's just him at home - disturb his coding day. Start getting the house sale ready. Wake up calls.
If he starts the job hunting process but can't secure a job in 6 weeks, ask your lender for a mortgage break to give him more time. But don't touch your savings. Let him blink first.
The therapy hasn't helped. You're keeping things together, but actually you're going to need to step up decisively at some point or things could go completely to shit.... Alternatively, he will slowly but surely suck the life out of you as countless men have done to countless women. He will exit middle age in good shape, you will not.
You don't have to be hostile, but I think you can say he's forced you into the position of being the only adult in the household so this is the way it going to have to be. He's making you his parents - his mother (nurturer) and father (provider). He is being neither. But parents don't just nurture and provide, they make the tough decisions. Don't dictate what kind of job he has to get, just the minimum amount that he needs to earn to make sure bills are covered. Keep reminding him that downsizing is another option. Basically treat him like a child, because that is the state he has entered - give him choices within parameters.
Even if he has a history of depression it sounds like becoming a dad - and you being on mat leave and him having to step up financially for a limited period - has triggered something in him. It would be good if you can still muster any kindness towards him (but wouldn't blame you if you can't) but be absolutely firm and consistent in what you are and are not prepared to do.
If the SAHD idea comes up I think you can bat that away with his ridiculous domestic efforts to date, so tell him he's not qualified for that job and you won't hire him. He'd only be suggesting it as a way to stay off work, not because he wants to care for his child.
Really sorry OP - what a let down.