Secretly (or not so secretly, based on looking at him), Mycroft has always longed to be a Victorian gentleman. He’d be humiliated to ever admit it, but he’s head over heels for the romance of the era. Hence the elaborate suits, the pocket watch (he practices with it in front of a mirror, to make sure he’s got that smooth time-checking motion down), and the brolly, which is the closest thing to a walking stick he can manage these days without either being laughed at or assumed to be disabled in some way.
I would take it further, exploring his internal fanboying of his own job. I mean: Parliament! Buckingham Palace! Unnecessarily elaborate politics and secrecy! He gets to cultivate a repressed Victorian air and nobody thinks it’s weird! He’s pretty much living the dream, as far as he can in this era.
Hm. Now that I think of it, it’d be kind of fun to turn it into a time travel thing, or maybe a sort of “Life as it looks in Mycroft’s head,” where he likes to picture Sherlock and John as our classic Holmes and Watson, living the Victorian bachelor lifestyle up in 221b. (And how thrilled is he that his brother moved in there? He has a silent fangasm every time he lays eyes on that wallpaper.)