Judging by the timeline established by the Gotham Times
, Begins and TDK are a little less than a year apart. The first issue of Times
has a story about a family dealing with a recent resurgence of the Fear Toxin only "weeks after" the initial attack. The second Times issue also mentions some event that will "run through the end of summer", so I'd say 6 months time between the two movies, with TDK taking place during the summertime, isn't unreasonable.
Now with TDKR there's this shot from the NYC shoot:
This is supposed to be like a promotional poster for a Gotham City museum exhibit. Despite the gorilla blocking our view, exhibits like this tend to run for a few months, so let's assume the dates are October 22, 2013 - March 12, 2014. They shot scenes with both artificial rain and snow, and there was plenty of on-location scenes that had neither, so the length of time TDKR takes place in is long enough to see changing seasons. Let's then assume the movie begins within the early run of this exhibition. TDKR in late 2013 - 8 = TDK taking place in late summer/early fall, Begins takes place late winter/early spring 2005...