I’m afraid I don’t know about qualifying for hours while you’re not working, but have you tried the Bubble app for babysitters? Lots available in London if you’re able to afford a couple of hours to let you job hunt. Hope you find a solution x
This happened to me too - lost my job whilst pregnant . It’s so so much harder than anyone realises as you are stuck trying to find a job without knowing how you will factor in childcare, you can’t get hours until you have a job, then you can’t get a nursery place until you know your finances! It’s really difficult and I have full sympathy with you xx
In Edinburgh there’s a crèche at the government building where anyone visiting can leave their child for a couple of hours and go work in the cafe. Have a look around at museums, government buildings, maybe an ikea, see if there’s a crèche available so you can get at least a couple of hours in fixing up your resume, searching and applying for jobs. Try asking in a group specifically for mums in London, they might have some recommendations.