In my opinion you can see noticeable improvement from changes in your training - like serious increase in mileage or introducing more and better speed work - after a time frame of at least 3 month, so it might be possible to run sub 20 in 5 month. 2.5 min sounds like a lot, but it is not unheard of for a beginning runner to have this kind of progression in the first few years. We are just talking about 22 to 20 min, not 17 to 15, which would be much more difficult to gain within half a year.
However, it won't be easy, either. Just imagine that the easy running pace for a sub 20 5 k runner is more like 8 min miles instead of your 9:30-9:45. You should be able to run your current 5 k pace of 7:15/mile for a full marathon distance then, too.
Maintaining 7:15 for a full marathon sounds incredibly overwhelming, as does 8 min mile being my easy pace (which would be sub tempo for me right now - doubtful if I could hold that pace for more than 40-50 minutes without strain). It has taken me 6 years to even get my weight down and get to a point where exercising for an hour or two a day is comfortable. How many minutes should I be aiming to run each day? I just ran for 55 minutes @ 9:35 pace.