According to this article it seems to have been a mix of Amey and Miller who were involved in the design and build, and Amey do the facilities management.
According to their (Amey's) website, they manage 10 primaries, 5 secondaries and 2 special schools (i.e. all the closed ones) covering a total of 15,000 pupils in Edinburgh.
this report has an appendix at the back (appendix B) which gives details of 6 large PFI contracts which were entered into around the same time and were used as case studies. Page 14 also gives a list of all of the first generation PFI deals for anyone who is interested.
As far as I can tell, Miller and Amey were also both involved in Glasgow and Fife PPP schools. It also looks like at least Miller were involved in Inverclyde - chances are most of the schools built in those regions around 2000ish will have been part of the PPI.
I think they've already been inspecting these schools but they'd been onsite at Oxgangs for weeks, had apparently thought on Thursday it was fine to open and then discovered something on Friday -linky to letter to council which says:
"shortly before noon today we were advised by the design and build contractor...they had discovered further serious defects as a result of which it has advised that these schools are no longer safe to occupy"
and goes on to confirm they'd given assurance on the 5th that they were safe.