I wouldn't worry about furnishings for now, that can be done over time.
We did an extension two years ago and it made a big difference to the living space, the family are much happier with all kids having their own rooms. It will be worth it.
Definitely see what you can sell in the short term, old clothes, toys etc.
Cut where you can on food costs, less meat, own brands, bulk cooking and freezing. Reduce outgoings on subscriptions if you have any. Think about the cost of all those mid week little shops, lunches out when at work/out with kids/friends, coffees, takeaways etc get rid of them, packed lunches and home cooked meals only.
No new things, clothes other than essentials like school uniform or going out places etc for a while.
Can you temporarily slightly reduce any direct debit payments then top up at a higher rate in a few months? Utilities sometimes let you do this. That might help if you have immediate costs now and know you can make it back up later. Make sure kids are turning off devices, lights etc and cut out long showers or baths to cut gas/electric costs.
Definitely consolidate debt to a 0% or lower interest payment as that will spiral quickly. Prioritise paying that down at more than the minimum payment.
As you save once the car and cat are sorted, put the rest of the money into a separate account to go towards furnishings otherwise it will end up spent on regular household costs. You could even set up a standing order each pay day to funnel it into a selection pot automatically.