Have you been in touch with the tilers, or was this DIY?
I have not used fix-a-floor but there should not be problems with the tiles over such a large area. If it were me I would do a small area and monitor before taking on the whole floor.
You did well to ditch the green roof. Unless the planners included a condition that the house could not be occupied until you have written approval for the roof as built there should not be an issue.
If you are on clay soil you would need to excavate a vast amount and backfill with the granular permeable base material. You may need to go down half a metre from the finish level.
I have laid permeable paving on a fairly thin bed of granite chippings on chalk subgrade and it drains very well.
The boreholes should not be too costly. If they are good designers they may use the information from the geotechnical report for a more cost effective solution for the pool.
Another vote for adding more battens. Just remove any that really mess up your gauge. The extra battens will strengthen the roof a bit. Could you get different shingles to work?
Only the timber cladding would be a concern, as well as the "composite" if it is flammable, especially if the breather paper and sheathing are also combustible.
For brands, Dewalt and Makita are good but cheaper brands like Erbauer and Titan can perform well. You will need more than one decent capacity battery and make sure it will fit any other tools you are considering. Go for 18V and batteries at least 3.0Ah. If you get stuff from Screwfix or Toolstation it is easy to return them if they fail.