James, your situation and age pretty much mirrors my son's. He too has a masters degree but has struggled with drink/drugs and anxiety since he was a teenager. Gaps in employment aren't that bad and on written applications you can be vague about them; it's only if/when you get an interview you might be quizzed a bit more. I'm not suggesting you lie but there are lots of ways of explaining gaps without being too specific ('health problems', travelling, etc.).
The other thing to remember is that the UK job market is quite good for employees at the moment and if you're not too fussy and/or happy to work via an agency there are a lot of jobs about. These agencies aren't too fussy about your history (some types of criminal convictions and/or inability to pass a DPS check excepted) and are more concerned with whether you'll stick it out long enough for them to get their fee. An example is parcel delivery - you'll walk into any number of multi drop parcel jobs is you have a driving licence. My son has done this a few times and now works in a plastics factory - boring shift work but relatively well paid.
Once you have a job and some employment history, moving onwards and upwards gets easier.
If you want any more detailed help feel free to PM me.