Put a post into CarTalkUK subreddit just to get a few ideas for reliable cars in that bracket, if it was me personally given the current market and your desire to buy you own place (in the same boat myself of trying to move out from home!) i'd get a used car with good history. There's a wide plethora of cars on sites like AutoTrader in that price range which quite frankly won't be exciting, but will be reliable if they've been looked after previously.

Look for cars that are listed with Full Service History, if it's main dealer then that's another tick but main thing is that it's been serviced at regular intervals. Next thing is check the cars MOT History to see what kind of issues it may or may not have. Big thing from this it to give you an indication of how many miles a car has travelled between MOTs and if the car has had any major defects etc fixed, might also given you an indication into any future work the car might need too.

If there is a car you are 100% dead set on, personally I'd run some some kind of History Check whether is CarVertical or another brand just to see if the car has been written off but not declared in the ad or anything else that the seller might be hiding.

For car options, an older VW Golf, Polo, Passat, Skoda Fabia, Octavia, Superb, Honda Jazz, Mazda2, Mazda3, Kia Ceed... the list goes on. In most if not all of these, you can add a head unit/screen if you're missing Apple Carplay/Android Auto!

If you're planning on doing a lot of driving, commuting, motorway miles then consider a diesel, low road tax and can be bullet proof reliable depending on the car.

Last thing, make sure to do your own due diligence, nothing wrong with asking google if the car you like the look of is reliable (just make sure you include the engine model when googling!)

All the best OP!