Because the shell variables inside a single-quoted string, they will not be expanded. It doesn't matter that the single-quoted string includes double-quotes inside of it, it is still a single-quoted string.
Use instead:
grep -a volume somefile | awk "/^$time1/,/^$time2/ {print}"
Actually, since print is the default action, one can simplify that to:
grep -a volume somefile | awk "/^$time1/,/^$time2/"
You do need to be careful with this approach. You must make sure that neither time1 nor time2 contain unintentional awk-active characters.
A safer approach is to transfer the shell variables to awk variables:
grep -a volume somefile | awk -v t1="$time1" -v t2="$time2" '($0~"^"t1),($0~"^"t2)'
Note that, even in this safer approach, as Gilles notes in the comments, you still must be careful: if the strings in time1 and time2 were to contain the characters ^*+.?\[|{$, the strings would be interpreted as regular expressions. This is not a problem for the strings shown in the question (they contain no regex-active characters) but, if other strings were to give you surprising results, this is a likely cause.
