I wouldn't call the LCY a major CTA!!!! (It must be one of the smallest in the world!)
If you mean transitting the LCY Zone then it is fairly easy - though a bit harder if going north to south.
If runway 28 is in use and
(a) You are a twin - then easily expect Alexandra Palace - south to Vauxhall-London Bridge
(b) A Single - then via the LCY overhead/Lea Valley (You can easily hold on the zone boundary at the southern most lake)
If runway 10 is in use
(a) Again as a twin you may get a route west of Canary Wharf (but subject to the LCY airport traffic
(b) Possibly the Lea Valley if there aren't too many arrivals or LCY overhead.
Be prepared to be asked to route via Alexandra Palace, Crystal Palace, the Lea Valley, The Dome, Vauxhall and London Bridges, the Airport overhead and the Thames Barrier.
If you are planning to stay outside, then why not follow the M25 (speaking to Stapleford at LAM on 122.8 for traffic info in the ATZ) Then stay on the M25 to Sevonoaks speaking to Biggin Approach on 129.4.
Of course you could be brave and take the Cook's tour via BUR-Ascot-OCK and then pick up the M25.
Good luck. Don't bother calling Thames if they are busy unless you are planning to transit the zone.