theres a good chance that the tiles/adhesive does contain asbestos BUT, the only way to be sure is to get it tested.

Now the good news. if the tiles are popping up easily you dont have anything to worry about, the asbestos fibers when becoming airbourne is when the concern should come into play, and thats only going to happen is if you start sanding the tiles, or the adhesive.

a putty knife or a long handle floor scrapper should pop up the rest of the tiles, if thats your goal. then you dont have to do anything with the adhesive if you dont want.

as far as the smell goes, you could try scrubbing with a bleach solution. theres all kinds of chemical cleaners that would work, or at least advertised to do so. if that doesnt work, you will need to get a professional over to take a peak and give you more solutions, its possible a polyurethane resin liquid mositure barrier might do the trick, but its expensive. I would personally try cleaning the concrete first before spending money.

Thanks Gregmills_gw. I just went down to check and all the tiles have curled and popped off effortlessly with a putty knife and the last bag of them has been removed! I strongly suspect the smell was coming from those tiles. I've just given the floor a good wash with Concorbium (? Sp) and we'll see how that deals with the smell. Strange thing is that the cloth came up black when I wiped the floor. Should adhesive with asbestos wash off like that? The floor is still very black but I found it odd that I picked it up so easily. I also wonder if it could be dry black mould that I picked up? The fun never ends......take my advice kiddies, never ever neglect your downspouts.
The humidity outside is at 50% so I've left one window open with my little room dehumidifier running at a setting so it won't shut off and I have an industrial air mover for circulation.
It may take a few days to completely dry then I can take full stock of the smell.

Given that I've just spent the last 3 days playing with asbestos tile and mold, I'm off to Google how to do a lung lavage at home. ( just kidding for those without a sense of humour)

My best guess from what you've described is that the adhesive is the emulsion type that contains no asbestos and not the solvent type that does. I worked with both tyoes, and if the wiping cloths came up black, it most likely is the emulsion type.

I had the same thing happen. Go to a place that sells commercial cleaning products. Buy a 5 gal bucket of gel orange cleaner. Spread it on with a long handled spreader. Let it sit 20-30minutes. Then take a garden hose and hose it down your floor drain. Yes, you can do this with asbestos in the glue in most areas (probably not CA). Most of the old black glue will come right up.