Well, that does help. Assuming your moving quite a bit at the gym, bending, lifting, running, etc. It almost rules out any on or over the ear. They simply do not stay in place and you would be constantly fiddling with them and of course the movement would cause sound leaks. Not to mention sweaty ear pads and headband.
Best bet would be to consider some nice comfortable iems. We still do not know what you will be using as a source or if you have an amp? I will assume you have a phone with no amp. Not sure of the quality of files you will listening to or whether you will be listening to lossy files from a streaming service.
I have been down the expensive iem road and am still ticked off when a $400 or $500 iem gets a snagged cable and boom, big repair bill. My advice would be a decent set of single driver iems as balanced armature are too flakey with impedance all over the map to say if it would reliable play well with your source, If you could to some test listening first that would be great. But a single dynamic dirver iem does not have the impedance issues which can dramatically change the freq response.
I use a pair of iems from Advance for my on the go walks. But since you are in the UK, why not look at something from RHA in Scotland. I also use their cheap pair as a backup and like them. Other brands that do well are Audio Technica, Shure and JVC which you should be able to source easily. Good luck!