Funnily enough I have just sat my asbestos awareness course. Potentially ever building in the UK pre-2000 has asbestos in it - floors, walls, insulation, cladding, roofs, paint coatings gaskets and seals to name a few places. There's constant asbestos fibres in the air we breath as that much was used in buildings.
However as long as it's not disturbed it doesn't pose a serious problem. I'm always wary in old boiler houses especially if the metal thieves have been stripping lagging to get to the metal pipes. Was recently in a hospital that the metal thieves has pulled every cable out the walls, floors and roof, the positive side if there was any asbestos in any of that they had also popped the water pipes so everything was saturated.
Personally I wouldn't be concerned at all, as long as you don't go throwing fibrous stuff around or kicking up huge amounts of dust. I know there is a lump of asbestos in the back wall of my house, from my window I can see 3 garages with asbestos roofs and know there is some in an old mill about 50 meters away. I also come across the stuff most days at work, HSE requirements to work in these areas is do not disturb the stuff.
A P3 paper mask is adequate, but how far do you go? If you are breathing fibres in they are also getting on your clothes, hair and any kit you have taken in. Strip off after an explore and put clothes in a sealed bag to dispose of, have a shower, needs to be done at the site otherwise you contaminate your car too, quarantine your equipment until an expert cleans it and says it safe to use.
Asbestos shouldn't be underestimated, however if you're not daft and avoid disturbing it I'm still happy to explore.
Clayton Hospital?