Do it now. You can’t really rely on the infill and it needs a proper slab. That will require work on both sides and if you leave it until the neighbour’s extension is built it will affect your new conservatory.
Sounds like you are on reasonable terms with the neighbour – maybe go halves on the cost? It would be worth making the slab suit the neighbour’s extension, eg, right level etc. Probably more a PITA for neighbour than you, given the lean-to and need to do spend money now, but by the same token, even more in their interests than yours.
I’m a civil engineer, btw.