Fishy ...... as Al says .......... good luck.
Building Control should be interested, especially if the type of material used has been changed. I'd bet my cheese sandwiches that they haven't considered the greatly increased loading that the roof structure now has to take with (I guess concrete) tiles.
Unfortunately this is often done. Conc. tiles are a lot cheaper and hugely quicker to lay than proper slate. These people are just being cheap!
Re-slating would let them easily tie into your roof and leave it unaffected. Tiles have no way of "knitting" to adjacent slate. What's usually used is a kind of leadwork "break" between the roofs. A slightly tortuous installation of lead soakers is probably the best way ......... IF done properly.
Another, older and less favoured method is to make up a "secret gutter" between the two materials ......... then cover with the tiles or the slates. The worst thing about this method is that gutters bung up .............. they're "secret"....... so you don't notice until water drips into the roofspace.
Good tradesmen have their own preferences, and would probably give me an argument about the best method, but anyone would agree this isn't the best way of re-roofing in a terrace.
Don't panic though ........... this is done all the time ............... it just needs a little more care to get it right.