Whilst there's a maximum weight used for airworthiness purposes, it's not a hard operational limit for most aircraft. It is a limit for microlights, as defined by various regulations, and will be placarded.
In light aeroplanes (which actually use a lower structural value than most microlights) it's not a legal limit - so long as the overall aircraft weight and balance are not exceeded.
Legally of course - any exceedence invalidates the CofA/Permit, insurance, etc. etc.
In reality, going overweight a bit, in the vast majority of smaller aeroplanes, the vast majority of the time, is not a problem.
Going outside of CG limits however, can get nasty very quickly.
Back to the OP's question - the answers given are good, but do phone to book, and ask about weight when you do. I agree, the Cessna 172 and Piper PA28 (variously called "Cherokee", "Warrior", "Archer".... depending upon exact model) should be absolutely fine. The C152 is a perfectly good aeroplane, but with somebody of 18 stone will be a bit limited and not many schools have ultra-lightweight instructors; the other common and popular 2-seat training aeroplane is the PA38 Tomahawk, about which I'd say the same.
G