I would go for taxi from LCY to Heathrow to be honest. I regularly drive from next door to LCY down the M4 to Slough at this time, departing at 7.30 gets me there at 9.00 on 9 days out of 10. Heathrow comes before Slough. It's actually one of the easiest journeys across London. If you can be out of LCY by 0745 (quite possible with hand baggage) you should be pulling up at Heathrow by 0915. And as you're hand baggage only and can do OLCI, should be OK.
I'm afraid the road/taxi journey to Paddington is worse for traffic and going by DLR/Underground/HEx takes significantly longer (typically over 2 hours from here).