Being a regular at LCY, although I've accordingly never done a transit there .....
You can be from aircraft steps to the immigration desks in 30 secs-2 minutes, depends on the gate but is v quick.
The IOs seem to work a sight more efficiently than those at Heathrow - you get the impression they can bang through the regulars in a flash but nothing untoward would ever get past them.
The baggage claim is right behind these desks and can be there within a few minutes of doors open - in fact you normally see baggage being unloaded before doors open.
From baggage claim there is no transit desk but it's just 30 seconds out of the door which leads straight to the check-in area/bag drop.
Up the escalator, through security at the top, and into the lounge.....
55 minutes; more achievable than a 3 hour transit at Heathrow I would say.