Well, I guess it's just time to be thankful that the underlayer is concrete. Those tiles will have to be removed - which wouldn't be a fun job if the underlay were floorboards.
As others here, I also believe the only way to do it will amount to brute force and a big hammer, chisel for the fine work. Judthepud2 knows what he's talking about.
I once lived in a flat with linoleum flooring (not tiles, just one big piece), and when I saw it, I thought - how cheap (I had been a spoiled wench with marble and under floor heating up until then). If I were having to pick out my own now, I would search high and low for that same linoleum. It was patterned like a sort of dark green granite (with no pretentions of fooling anyone into thinking it was stone), so it didn't show every speck of dust as soon as it hit the floor. It also had a softer underlay - much easier on the joints - and 'drop' didn't unequivocally mean 'destroyed'. I would also never have underfloor heating put in again, as it made my ankles swell.
Good luck with whatever you decide on, and I hope you love it!