If you are reasonably quick in responding to the e-mails that you receive from the TSA, I think the whole process should not last more than a few days. There are several steps to the process, but on the TSA side they're all automated, at least until the point they receive your fingerprints. So until then, it won't be the TSA holding things up.
The background check that the TSA does after receiving your fingerprints is more involved, but the funny thing is, you may already start training after the TSA acknowledges receipt of your fingerprints, *before* the background check is complete. At least, that applies for private pilots. For more advanced training you need to wait until the background check is done.
If you find a place that does electronic fingerprinting (check the TSA site) you should be able to have the TSA send you acknowledgment of receipt of your fingerprints (which is the magic e-mail that allows you to start training) within two or three days, I guess.
Since your primary purpose of being in the US is not flight training I would say that you do not need an M-1 visa for this. In fact, you might already have another visa as a permanent resident and I don't think you can hold two visas simultaneously. However, as with all things visa, I would contact the relevant authorities and ask to make sure. And read the fine print that comes with your current visa.