You can get an M1 with a WWII era receiver from the CMP. You can also get an "as issued" M1 from the CMP. You can not get an all correct, original, as manufactured M1 from the CMP....well maybe you could, but these rifles are few and far between and when the CMP finds them they would be sold at auction for much more than the service grade price.
If you want a bona fide all correct WWII rifle your best bet would be to buy from a respected dealer. Scott Duff is one of the best and he has a nice, all correct, late war Springfield Armory (SA) M1 for sale on his
site right now. Looking at what Duff has will also give you an idea of how much such rifles are going for now.
If an M1 with a WWII era receiver will make you happy then you can order a Winchester from the CMP (one to two year wait). It will have been rebuilt and have parts from different manufacturers and eras, but will be an "as issued" USGI M1. You could try asking for an SA with a WWII era serial number but the CMP really dosen't like such requests; they are more work for them and its not fair to other buyers. The Danish rifles were given to the Danes from stocks leftover from WWII and used by our troops first so, if the Danes are still available, you could take a chance on one of those.
A trip to CMP North to pickout your own is also a good idea.
The Greeks got mostly new post war rifles. I saw crates and crates of the Greek rifles at the CMP just after they arrived and there were
very few WWII era rifles in them.
Just for fun here's a pic of the Greeks in the CMP warehouse just after they came in:
HTH