Irrespective of whether or not the wrong people read this site, do you all really think the legality of flying single engined over London or anywhere else would worry the type of people you are talking about? You're not ALLOWED to fly passenger jets into skyscrapers deliberately, but they did.
So please stop this tabloid-style sensationalism.
The original question was a perfectly valid one. The report about the LCY CTR being 24H is news to me, but I am happy to accept it. Certainly I used to fly a twin regularly on a Sunday morning over London and was told by Thames that the CTR was not active. (It often became active at 1300). I have on more than one occasion done a practice ILS into LCY in the open FIR (having told Thames that I was doing it) and the only thing to worry about is Rule 5 on the climbout.
The issue of doing it in a single-engined is one of those grey areas which can be debated all week. My feeling is that at 2400 ft you have about 5 nm glide distance in still wind, which is enough to make it into the Lea Valley or LCY or the open ground south of the river. It all depends upon your assessment of the meaning of 'alight clear'. If you take it to mean clear of the whole conglomeration of London, then you have no chance. If you take it to mean clear of a congested area (e.g. into the Lea Valley) then you are OK. Your licence, your choice.
Thames can indeed be very helpful, but bear in mind that it is not ATC's responsibility to ensure you are complying with the law. If they send you via Westminster to avoid traffic and you are single engined, then I suggest you request another route!