Old tiles - clean or remove?

Hi

In the bathroom of my 1917 house, the original black and white tessellated tiles have a very poor quality surface to them, due to wear and tear over the years. I think that perhaps when they were scraped of all the 'lovely' dental blue green 60's paint off them, this might have taken off a fine layer of tile off the top. There are also old pink paint stains that I just cannot get rid of, no matter what product I use. I think my main question is, is there any feasible cleaning product that would bring back the shine of the tiles aswell as removing any stains that have blemished the floor over the years.. OR, is it worth getting rid of all the tiles, perhaps improving drainage of the concrete base underneath, putting floor heating in (very cold bathroom - double brick house), and purchasing new reproduction tiles which fit the original pattern...?

The only reason I'm think of the latter, is I'm going to be moving the position of the bath/shower/toilet around...