With only five locations or so in Europe, and no agreement in place between the State Department and the Department of Homeland Security to get printed at the embassy/consulate (*), it will be virtually impossible to get fingerprinted before you travel to the US. So you need to go to a Sheriffs office as soon as you arrive in the US (check they're open), get fingerprinted there, submit the documents, hang around until the TSA acknowledges receipt (might take up to five days) and only then start flight training.
When I was there the school, at that time, never awaited the TSA "receipt" mail but sent me in the air straight away. Although I did not solo yet at that time. But with the new spot checks of the TSA (as reported by SoCal), the flight school might just stick to the rules and not allow you to receive training until the TSA has said so. Which, for a cat 3 alien, is at the point where the TSA acknowledges reception of the fingerprints.
(*) The ridiculous thing is that every US embassy/consulate has fingerprinting equipment, because they require your fingerprint for your visa! But the TSA doesn't accept these.
For a cat 1 alien (flight training above 12.500 lbs MTOW) it's even worse. You need to hang around waiting for the TSA to actually clear you before you can initiate training. This can take up to 30 days.
The whole TSA thing is very easy. It's just that you cannot get fingerprinted, for all practical purposes, before you get to the US.