Thanks BackPacker for the clarification. I had based my general statement on an email that I received from the flying school that had the following in it;
"Please note that training cannot begin until the TSA grants 'Permission to Train' – please plan 3-7 working days to receive this, and note this time is NOT included in the course duration".
I figured I would split the difference and assume 5 days. However, my use of the word approved was probably inappropriate as it is really the acknowledgement of receipt that matters, as you pointed out.
It certainly seems that if the fingerprints I submitted from Europe are rejected because they are not of a high enough quality that I can find a collection office in the US when I get there that has the electronic live scan facilities and try again. Hopefully it doesn't come to that though.