I had a PCP car with no deposit, it was 0% finance so made no difference to what it cost and I was actually planning to pay cash, but there was no discount available for this and by taking finance I got free servicing and breakdown cover, which wasn't available if I paid cash, so worked out better that way. 
Are there not any nearly new cars available in dealers OP? I also had to get a new car in an emergency a few weeks ago as my old one failed its MOT and wasn't worth repairing.
I was worried about the shortage but when I started looking there was a few in dealers that were what I was looking for, but I saw one that was advertised on ebay by a car supermarket. I looked to see what was available of this type and it was actually a reasonable deal - a 2 YO Skoda Citigo for £8k. I was initially going to get one a little older, but it wasn't much more for this one. It was also at a branch that was only a couple of miles from my house so easy to view and collect.
Mine came from Available Car which has a few branches in northern England/Midlands, but there are a few similar outfits across the country. I also looked at Cinch/Cazoo but they were more expensive.
I test drove it within a day and picked it up after about a week, and it would have been quicker apart from it had only just come in and needed checking and preparing for sale.
Interestingly, I've just looked at their website to see what they have of this type of car right now and there's only 4 Citigos instead of 10 when I was looking and one I looked at but hasn't sold has now actually increased in price by about £500. The closest comparison to the one I bought is a year older, higher mileage and nearly £1k more expensive.