UK in May, correct? (The where and when are crucial bits of info...also, the activities. City walking different from hiking. Etc.)
Anyway...if I were heading to the UK in May (and how I wish I were!!!) I would bring one pair of Ecco Soft sneakers (white or silver, with slim Superfeet insole because I can't wear them otherwise.) Mr. Suz and I would probably be heading out for meals so I would also want a slightly dressier shoe -- some type of dressy-ish flat or low heel. It is awfully hard for me to find this type of footwear though, and I don't currently own such a shoe, unless you count my chunky loafers, which don't seem dressy to me -- so I am not sure where I'd look or what I'd seek. Possibly something from Eileen Fisher, since her shoes have worked for me. Possibly something at Ecco or a Franco Sarto. Possibly Sofft, though those are harder to source in Canada.
If anticipating rain then I would probably want a darker shoe also, although I wore my silver Eccos in POURING rainstorms last May in Ontario (real deluges, walking in them for ages) and although they got awfully squelchy and took a while to dry out, they weathered it all pretty darn well. But I think the possibility of serious rain means you need to have more than one pair of shoes with you, for sure.
ETA if travelling in cooler weather, I would definitely bring/ wear a boot. If travelling in hot weather, sandals for sure! In my case, probably my EF's.