Not viable, due to signal sighting as the driver would pretty much be stopped just short of the tunnel and blocking the throat of Kings Cross.
On the contrary, quite viable. If it was decided that the best option for at least one of the sleepers was to have them departing from Kings Cross with split trains boarding at multiple platforms, a solution to make it work would be found. Other nations all over the world manage to handle long trains having boarding split over multiple platforms or the addition of car carriers. Surely the mighty railway system of Britain would be capable of sending some of their top brains on a research trip to find out how everybody else manages it. As they say,
where there's a will there's a way.
If the two portions were both EMUs, the front portion could proceed to a point where the rear vehicle was clear of the points leading to the other platform, after which the other portion drives forward to couple. Scharfenberg couplings would make the process quick enough that it would only be a difference of a minute or two compared to having consecutive departures of two shorter trains. If they were coaches then it would be almost the same, with the loco which hauled the rear portion into the station pushing the coaches out to couple with the front portion, a process that could be even easier if the sleeper coaches were set up for push-pull with a cab at each end of the rake.
All of it would be done using the two platforms farthest across to one side of the station, and the corresponding track, so that only part of the station (one track in and only a couple of platforms other than those in use by the sleeper) is blocked and only for about five minutes.
How feasible would it be to start as two portions with one departing slightly ahead of the other, and joining at somewhere like Finsbury Park?
This is the point where it would start to become impractical. If you did it just outside the station it would be a significantly more efficient process because it's just shunting moves controlled locally. If you did it at some station further down the line you've got to have headway between the two trains and then the process of coupling the two portions.
If they were that bothered about problems in London they'd terminate it at Watford (which is clearly a silly idea).
If they were that bothered about problems in London they would hopefully get their s**t together and sort things out so all the other non-sleeper trains can run properly too!