Good point. The Old Vic has 7m each side, the Phoenix 2.74 / 2.13.
From what I remember, there wasn't much trundling on and off, so I'm guessing it could work?
I think the 3 pieces that created Phil's bedroom were probably about 2-3m each and then theres the bar, bandstand and all the rest, before prop tables and quick change areas come into play. Wing space would definitely be an issue at the Phoenix.
As a completely uninformed guess, I recon the Broadway show got pulled as a result of an investor pulling out, expecting a quicker return and not being happy. Thus leaving the show short and finances collapsing. I expect the same investor had initially agreed to fund the US tour, but after pulling out of the Broadway run pulled from the tour and their financial commitment couldn't be replaced. Hopefully this is less of an issue with regard to a UK installation as I think the nature of investing and funding productions here is quite different to the US. Just a theory. If the UK run doesn't happen, Minchin will have had a real bad time lately. With his movie getting pulled after 4(?) years of work and GHD US bailing at a loss. Do hope a UK run can help turn it around.