Remember if you go for a second hand cheap car, you will need to put aside money for repairs, which can be significant, no car to tied you over, plus if the engine goes, you are looking at £5,000 for replacement and fitting for example.
I wouldn’t pay 5k for a new to me car, let alone for a new engine. I’d just scrap the car and get another for £500.
I ran a £461 Ebay purchase for over 6 years and the upkeep/repairs/MOT averaged out at a couple of hundred at most a year. My last car before Motability cost me £750 and it cost me the grand total of £10 in repairs in almost 3 years (a dirty sensor), after that it was just the MOT fee which it sailed through each time and a couple of tyres and some oil.
I’ve always had a courtesy car from the garage doing repairs or an MOT.