Renovating Victorian fireplace surround - what do I need?

[dogseggs]

I wonder if anyone can help? Our Victorian fireplace is in need of some tlc. The surround and mantelpiece are made of black marble (effect?) stone and it's all looking worn and tatty.
The mantelpiece itself is wearing to a dull grey at the edges, and the upright surrounds are covered in a yellowing varnish which I have tried scrubbing off with no luck (see pics)
Two questions:
is there a gentle varnish remover I can use to clean it off without damaging the stone?
and is there some sort of wax/paint I can use to brighten up the whole thing?
We still use the fire occasionally so it would need to be heat proof.
Any advice gratefully received!

It will be slate, probably painted to look like marble. Paint stripper won't harm it. Finish it with oil.

Thanks, it may well be slate. Wouldn't paint stripper take the white marble effect paint off as well?

It will take all the paint off (I'd got it into my head you wanted rid of it all), but I doubt you'd be able to just get the varnish off. Maybe one of those razor blade scrappers to get the flakey bits off and a fresh coat of varnish?
If it goes wrong you can always fully strip it and go with plan b (oiled slate) or c (marbling it yourself)