Upvote:2
They need one Schengen visa for Italy and Norway, or two separate ones. If they visit one directly after the other, they need one visa; if they return to the UK, they need either a double-entry visa or two separate ones.
You also need an UK visa, which is a completely separate process. However, it can be easier to get a visa for places like the UK or Schengen area if one has a visa in another Western industrialized country; if you return to the UK from Italy/Norway, it helps if you already have an UK visa.
I can't answer average processing times in the Philippines.