Pubrunner might be able to shed some light on it, but I've not noticed him around for a while.
Pubrunner's baffled, I'm afraid !
From the pics, what I can say, is that it looks like a very good quality frame - I think, from the (early)50s. All that I can suggest, is that a similar query is placed on the Retrobike. It might be worth joining the V-CC, to see if you can find out anything more - if you wish (pm me if you do), I could contact the Marque specialist for the London builders, to see if he can tell (me) you anything more. Another source of info., might be the cycle museum at Llandrindod Wells.
Since you know of the (approx.) area of London where the frame is from, it might be worth contacting a local newspaper, to see if they would run a feature on it - asking for information.
Of course, it is quite possible (likely) that Toledo Cycles were merely the retailer and not the builder of the frame.
If it were mine, I'd do everything to retain the bike's identity - it is obviously
very rare indeed. It is the kind of bike that would garner a considerable amount of interest at an old bike meet. It is a 'mystery' bike, but I think that you'll have a lot of fun, that trying to solve the puzzle.