Hi.
If you want to record in a studio you can check out Pirate. They have studios across London.
Their studios have 2xPioneer CDJ2000 and a Pioneer mixer ready to go and the recording process is super easy. All you need is a USB of digital tunes to take with you. It is ~£35 for a couple of hours. Although it’s not much use unless your music is digital.
In terms of recording a mix, or ripping vinyl, Audacity is free and easy to use. There’s a whole thread on here about digital DJing with lots of recommendations for devices to connect your analog mixer to your laptop. I use an iRig Stream which is pretty cheap and effective.
But you also say you’ve only got one deck.
If your main source of music is on vinyl, personally I’d take my laptop and iRig to a mate with two decks and a mixer and record my set that way. Or if you aren’t beat mixing you can just record one tune after the other into Audacity and edit out the gaps afterwards.