How serious is building subsidence
The very likely outcome will be that the structural engineer will go away to do the necessary calculations, specify the requirement for the underpinning to be carried out by a contractor, and the tree will be converted to firewood (tree preservation order permitting).

The underpinning will be carried out and leave the building stronger than it was before. Any additional remedial work (eg to the cracks) will be carried out. Over time (once the repairs have weathered), you probably won't even be able to tell that the work was done.

The problem has been identified before the worst case has happened (crater), so now it can be fixed to mark sure that it doesn't happen.

Without a thorough assessment by a qualified individual (or company), I can't say for absolute certain, but I'd say it is very, very likely that a satisfactory solution to remedy the subsidence will be found.