Basic training during WWII was 10 weeks. However it involved a lot of rifle cleaning, marching and running that Steve wouldn't have needed.
Read about Camp X,
the training camp for OSS (pre CIA) agents, training there was 1 month in duration. Ian Flemming was trained there and it's claimed he based James Bond on Sir William Stephenson whom he met there.
The scene in the movie where Steve is designing his uniform from Howard's carbon fibre and getting the vibranium shield, we can presume that there was few weeks there of preparation where he could be trained, possibly by Peggy.
Let's also not forget Steve was always a scrapper, he got beat up because he didn't have the muscle to back up his moves, that doesn't mean he didn't pick up any experience.
However I agree, overall a glimpse or a reference to him getting some hand-to-hand training would have been nice, and taken about 12 seconds.