Lots going on here...
Selling the house isn't an ideal back up plan for when the money runs out as the housing market is slow and the process could take a year or so.
Remortgaging - you're taking a massive risk doing this and not being honest about the job loss.
11K - even with a 2K mortgage, that should last at least 3 months if you live frugally.
Crazy that you both didn't have at least a 6 month emergency fund when he was earning £80K.
I think you are being harsh to have expected him to have had interviews within 4 weeks. The job market is very slow and harsh right now. My DH's colleagues were let go 2 months ago and both are still out of work, one has had one interview. I think you need to be prepared for him being out of his 'normal' work for at least a year. I say 'normal' because he will need to get ANY type of work at some point soon.
Now for your DH, I think the first couple of weeks, he can be cut some slack, I've lost a job myself and know how it feels. However, my sympathy has run out for him after that when he's refusing to do childcare, pick ups/drop offs and housework. Surely even when he was working, he was doing his share? No?
He should be spending a good 3-4hours per day job hunting and applying for roles, that's seriously searching, adjusting CV, applying, not scrolling the internet and getting distracted. If he's not spending 3-4hrs daily as a minimum, then he's taking the piss.
He needs to stop with the days out meeting friends or whatever, you can't bloody afford it right now.
It really does sound like he's getting too comfortable not working, and knowing you're going back to work, I think he's got it in mind that you are so worried that you'll go back full-time, and he can do what the fuck he wants except for the housework and childcare.
He needs to be signed on for JSA to keep up with his NI for his state pension.
You both also need to discuss at what stage he starts applying for ANY and ALL jobs. You cannot be choosy in this current climate.
You both also need to discuss him taking on childcare and housework, not necessarily all, but if you go back full-time whilst he's unemployed, he needs to be doing the bulk of it all.
You both also need to discuss ditching the nursery and him looking after the children, it's an expense you cannot afford right now. You may have to give a term's notice, so this could be something to seriously consider.
I think you need to sit him down and make it very clear the severity of the situation, and that he has NO excuse not to be doing housework and childcare whilst also job hunting.
Frankly, he's taking the fucking piss.