I'm sure you've been to Kew Gardens, Carole, but if not that's top of the list.
If you've not been there before, the Chelsea Physic Garden is interesting, especially as its unusual city-centre Thames-side microclimate allows it to grow a lot of mediterranean-climate plants. And they usually have their own seeds for sale, some of which are interesting (our Koelreuteria paniculata mother tree in France, which now has offspring elsewhere in Europe, came from seed from their fine old tree, one of the first to be grown in Europe).
If you want to immerse yourself in gardening books and magazines, the best place is the Lindley Library of the Royal Horticultural Society.
Helena and I are an hour by train from Charing Cross, and you'd be very welcome, but it would be more a matter of us enjoying your visit as our garden's looking rather bedraggled now! (Some pretty autumn colour though, and two dozen different sorts of pumpkin/winter squash, as well as quite a few different chilli varieties.)