I know that using the scan on the galaxy map will attract reaper attention, but is it also true for the planet part (where you launch the probe)? Basically do I have to rush this bit or can I take my time when finding all the loot?
Scanning planets does not attract Reaper attention. You can take as much time as you want.
You can even scan planets after Reapers start chasing you. Everything in the star system is frozen when you click "Enter Orbit".
The problem is that in order to scan a planet or debris field, the system itself now needs to be scanned for triggers first, which does attract Reaper attention according to the message displayed after first entering a system.