Thanks, everyone - I'm not in London so can't use those London-specific options.
The things I need to store aren't jewellery etc. but documents and data. I'm aware of cloud storage, of course, but have concerns over security/hacking etc.
I'd like to be able to keep back-up copies outside my home but would need easy access so that I could keep storing updated copies.
I live in a city but don't have a car and have mobility problems so I'm really looking for somewhere that would be on a High Street rather than on some industrial park that I couldn't get to (so not large storage units).
I live in a flat and don't have a garage or car that I could store things in (my main concern is fire).
This Metrosafe article suggests that HSBC no longer offer safety deposit boxes:
http://www.metrosafe.co.uk/editorials/news-releases/date/2015/12/09/what-replaces-hsbc-and-barclays-after-uk-exit-from-safe-deposit-boxes/
Interesting idea about asking the local police!
And good advice about giving access details to a trusted person.