Just get cheap rugs or offcuts to cover as much of the floor as possible and then do carpet or vinyl as you can afford it. Amazon has many very cheap rugs of different sizes.
From experience, you need to prioritise in this order;
Kitchen/bathroom vinyl if there isn't already some or vinyl tiles in place - usually very cheap because the spaces are very small. Sticky tiles are OK if they're on sale in the poundshop (it's where I got some for my old flat's bathroom), but they're not as durable as a sheet of vinyl.
A covering on the ground floor to make the most difference in winter - definitely add in some thermal underlay, anything is better than nothing. So hallway and living room. Depending upon the size, the hall can usually be done with a very cheap offcut and you just make do with the largest area you can covered in the living room.
Rugs by each bed.
Stairs - whilst bare treads can be fashionable in warmer homes, that's unlikely with most social housing and stacking on them hurts. Depending upon the width, it could be more realistic to buy three cheap runners and then fix them with carpet tacks than expect to fully fit carpet at first, possibly if they're quite wide, painting either side will mean it looks better.
Landing - could be another offcut tacked down.
That's enough to keep you all warm enough until you can gradually get together the funds to have the floors done properly over the coming months/years.
And practically, if you're bothered about fitting vinyl, I found that giving the edges a blast with a hairdryer made them soft enough to cut through easily and neatly right up against the skirting with a Stanley blade.