Harlequins and Saracens have amateur sides at the next level down L10 and Wasps had an amateur side too but not sure where they are now.
The interesting ones are London Irish Wild Geese (always been the LI second side and a real power in the 1970s even with B52 - in those days they, Quins Wanderers, Richmond Saxons, London Scots Picts, London Welsh Druids were very good sides and regularly turning out internationals in their teams.
Over the last decade Wild Geese have had access to the original and now the new Professional training grounds in Sunbury, some coaching support, and some Academy / AASE St Paul's College players. They are currently top of National 3 (London & SE) and were up in National 2 last season. Originally called London Irish Amateurs, I reffed them at L7 in Hampshire around 2007/08 as they were climbing up the leagues,mother renamed as Wild Geese three seasons ago and are a separate financially entity to the professional club.
London Irish Geese are LOT more connected to London Irish than Harelquins Amateurs are to quins (which is hardly connected at all)
I think this reflects the heritage of the clubs, pace the Wanderers, my impression is that Quins were traditionally a club only really interested in the 1st XV, while LI were always a more community organisation.