What are peoples experiences about booking the cheap London Spezial tickets ?
If I understood correctly you can get them to most German cities, but I've never had much luck getting London-Stuttgart at a low price. Are they restricted in numbers or to certain routes or to off peak services (making it less likely to get longer journeys of multiple legs)?
Like all German Advance tickets (Sparpreis, Europa-Spezial, London-Spezial), they're subject to yield management, so the price tends to go up as the train fills up (or as the Advance ticket quota for a given price level gets sold out), so booking further ahead can get cheaper.
However, I believe they can choose to "lock out" one or more of the cheapest levels directly, so for a busy time of day or day of week may find that you don't get the cheapest price even if you book a minute after midnight German time, simply because that price isn't available for that journey.
Also, tickets that allow travel by ICE are generally more expensive than those that do not allow this.
Though I'm not sure how relevant this last condition is for London Spezial, because one of the conditions is that you must go from Brussels to Cologne by ICE, so it's a "ticket allowing travel by ICE" by definition. (I think that ticket price is not affected by how much of the trip is on an ICE, just whether ICE is allowed at all for any part of the ride.)
There is no "peak" or "off-peak" time per se on German rail.
Prices are generally set point-to-point, i.e. not by mileage or the like. It can be cheaper to get a ticket for a smaller station that's a bit further away. (Note that stopping short is permitted in Germany -- as is starting short, but your seat reservation (if any) may have expired since it's only held for you for 15 minutes after the train leaves the station you had reserved from.)