First off... "nail sickness" with iron nails would suggest that this is either the original roof, or that the roof was re-nailed a long time ago.
TWR mentions "ripping" them off. This would be the case with a more recent re-nail, as the nails would be in better condition. I would guess that, in your case, the slates should be removed starting from the top. Each slate should be carefully lifted off. Either the nails will finally break, or the hole in the slate would enlarge to allow the head of the nail to pass through.
Either way, Welsh slates are among the best (along with Cornish Delabole.) In good condition they can be worth several pounds each. Only a close inspection would tell if they are worth this amount. After 130 years, they may well be what is known in the business as a "bag of biscuits".
I'm intrigued to know what the "brass studs" are. You say concrete, but are they possibly modern thin "fibre-cement" slates? (Used to be called Asbestos cement.) Fibre cement are nailed and also held at the bottom of the slate with a copper "crampon" which is bent over to hold them down.
Although modern, synthetic, slates/tiles work very well, they can look dreadful in comparison to a traditional, natural slate. What about having a complete re-nail using your originals plus a load of reclaimed bought to make up for inevitable breakages when removing them. This is done all the time.
Any roofer who baulks at re-nailing should be avoided (unless it's a lot cheaper of course.) A proper, traditional roofer would take great pride in covering with reclaimed slate. Others just wouldn't want the bother. It's so easy to tear off the old ones and quickly fit modern equivalents because they're so easy to lay.