I agree that if there are any direct debits/payments you could cancel in January (fun things like netflix, not essentials obviously), then it might be an idea to stop them for the month, even if it's small amounts it may help stop you going overdrawn by accident. There may also be things that you could change from monthly payments to quarterly payments- so you could pay in March instead- obviously you'd have to remember to save for these, but it would hopefully help in the short term.
If you have two weeks off in December, and it would be possible, you could see if there are any businesses that might want casual help for those two weeks- try pubs and restaurants rather than big companies. This could bring you in a little bit for the two weeks when you aren't earning anything. I would also explain the situation to family and friends, and maybe they will be happy to host you for meals etc. during this period which could help reduce your outgoings, and you could return the favour in the new year. You could also have a pre-Christmas clear out and see if there is anything you could sell.
It sounds like the new job will give you much better prospects that your old one, but I think a lot of people would struggle with no money coming in for nearly 6 weeks. If you think you will have problems paying major bills such as rent/mortgage, it might be an idea to talk to your Landlord or bank in advance and explain the situation. Banks especially do sometimes have options they can offer to reduce your payments temporarily.
If things are going to be very tight, it might be an idea to agree an overdraft with your bank now- this should help you save on fees etc if you do go overdrawn, obviously it's not ideal to get in debt, but you will hopefully be able to pay it off when you are earning more. This would also potentially cover you if an unexpected expense came up in a January.
In the next four weeks I would save as much as you can so you have a buffer and budget everything as tightly as you can in January, really look at your outgoings and see if there are places you can make big cuts- if it's just for a month, hopefully you/your family can endure some short term pain?
Will you have any income e.g. child benefit in January?