It would be wonderful to be able to combine the musical with the concert in London.
But given that MK usually plays London at the end of May or the beginnging of June, expectations should not be too high.
And maybe it is even better so, since almost every fan who plans to come to London for the show would also wish a ticket for the musical during his stay and that should prove more than difficult.
Yes, MK usually plays in London late May, early June as later (that was in 2001, STP tour if memory serves).
This time I think it could be different because of the musical. MK would be of course in the premiere in Edinburgh, and I believe he would also be in the premiere in London, which is June 13rd, or earlier if that's the commercial first date and there is an earlier premiere (as it happens in Edinburgh), so it's likely that MK would:
a/ Place the London dates around the London premiere for the musical so he can attend as the writter of the songs
b/ Place the tour dates with a gap of a few days so he can travel to London to attend the premiere, and then go back to his tour
Both are logical moves, but I think that most of us would preffer the first one, so we can do a double!
By the way, The Old Vic gives the chances to buy very cheap tickets to previews of the plays... maybe there would be five pre.view shows before the one at June 13rd that coincide with him playing in London! Check this:
https://www.oldvictheatre.com/your-visit/booking-tickets/pwc