I'd get some sort of doctor's note. I don't know how legally compelling they are/can be, I'm not sure if/to what extent anxiety may be covered by the Disability Discrimination Act. fortunately for me epilepsy is. In my days working at the Tesco I worked staggered days on nights. After a period of doing so it wasn't having the greatest effect on me (and some other things which weren't really relevant but kinda helped me with my case) so I had the neurology doctor I go see to write a formal letter saying he recommends I go back on days. There was little delay in them moving me, but they had to create the hours as there wasn't an actual vacancy available. It wasn't incredibly ideal but was an improvement on what I had (although the decrease in pay was a bit of a ballache). I think had they refused I could've tied their anuses up with so much litigation, their grandchildren were gonna need lawyers #PhillipBanks.
In my current job though after having my probation extended it appeared I had made a few errors, which could've been quite serious, the silver lining for me is that they happened before the probation meeting, so it couldn't be said I hadn't improved since the meeting, but you think you're having a productive day and then Pow - you find out you fucked something up and have to get your anus in gear to rectify things tootsweet. It's lead to several meetings and a lot of patience from my supervisor and me thinking of several ways to say "I messed up" in the most positive light. So they want me to work on some better reference guides/personal checklists that I am about to work on to show more accountability and confidence in things being done.