If the plane sits overnight at LCY it might leave. The minimums for take off seem less stringent for take offs than for landings. If your plane is one that first comes in on the first wave of arrivals, I'd be less sure, as early arrival diversions are fairly common. The fog (usually) burns off as soon as the sun is properly up.
This is none scientific observation, not inside knowledge, so happy to be corrected by an insider.