Great ideas above. There are some sandstone outcrops (like Harrison Rocks near Tunbridge Wells) but at the top you aren't much higher than surrounding hills so that might not convince you've climbed a mountain! At least the top of Leith Hill has a tower marking it out, and an expansive view. But most of the geology near London gives quite rounded tops. How far do you want to drive / how close does it need to be and from which bit of London? Places like Malvern Hills rise really dramatically so feel like a mountain, about 3 hours drive I guess, or there are many moderate peaks in the Peak District that would work for a camping trip.
“Or get out to the Chilterns, but I couldnt give you a decent hill there just the area.”
In the Chilterns you could try Ivinghoe Beacon.
Leith Hill is outstanding. It's woodland walks rather than scrambling, but I think it's the highest point in SE England. From the top in one direction you can see the skyscrapers of the City and Canary Wharf. In the other direction you can see huge distances across Surrey and Sussex, watch planes taking off from Gatwick, and apparently on a clear day get a glimpse of the Channel (not quite when we went).
If you go after lockdown when Leith Hill Tower reopens, all the better.