For the Great Britain, the Ordnance Survey National Grid (and in Ireland, their grid) I would regard as the best system to use - it's marked on OS and many other maps, as well as being accessible via various apps (from th OS and others).
MGRS is the global equivalent, used by the military for the rest of the world - and it is, to all intents and purposes, the same as USNG (United States National Grid) - there are apps/web sites which give locations and maps, which tend not to restrict themselves to the USA. Within the US, one of the adoption messages for it is that, when you call in the Feds, that's what what they're going to use, so adopt and get used to it! But outside the military and the USA, I don't know how many people know about them.
But one problem with MGRS/USNG is that - although global - it is divided into zones, with discontinuities at zone boundaries.
Most other countries have their own survey grids, although I've no idea to what extent these are used by the public.