Which instruments typically double up (play unison) in military bands? I have googled this question in different ways, but without an answer.
This would be purely up to the composer/arranger and the particular sound they want. Short of a rule to follow, the best option is to do a study of orchestration of martial music. That study could be done primarily in two ways.
It depends on the size of the band. Some bands are so small that there is one player per part (maybe not even that, some parts might not be covered). In the case of a huge band almost everything might be doubled (or tripled, or more). In general, any time there are more players of an instrument than there are parts for it, some of the parts will be doubled. Mostly the first parts are doubled, but in some cases it can make more sense to double a lower part (e.g. if the first player is very strong and all the others much weaker).
For average sized bands (30-40 piece) you can generally rely on each of the B-flat clarinet and flute parts to be doubled and maybe the first trumpet/cornet.