There's only one security queue at LCY, so flying business or having status doesn't make any difference. There are up to 5 scanning stations, and they do tend to open all of them at busy times. Most people travelling through City Airport are regular travellers, so know how to prepare for security, and consequently things tend to move pretty quickly.
I would say that 20 minutes is cutting it rather fine, especially if you don't know the airport well. You'll need to allow 2-3 minutes from DLR to checkin, then a minute or two to print a boarding pass (longer if checking luggage), then 1-2 minutes minimum from checkin upstairs to security. Even assuming no-one ahead of you, you've then got probably 2 minutes to unpack / security / wait for bags / repack, then walk to the gate.
It's not unheard of for a BA flight to begin boarding before checkin closes, so if you cut it that fine you may struggle to find somewhere nearby in the overhead racks!
If you're after anecdotes, you'll find a lot in these two FlyerTalk BAEC threads on LCY security. Personally, I'd not want to cut it finer than 30 minutes at LCY. There are decent places to eat and drink before and after security, and free wifi throughout the airport, so getting in with a bit more time to spare isn't the end of the world. If you're aiming for 20 minutes and the DLR has a tiny hiccup you're stuffed, so best not risk it!