Mycroft might be a naturist
While the whole scene is made for comedic effect (aka. "rule of funny"), there is a possible explanation: Mycroft might be a naturist, enjoying non-sexual nudity.
The Wikipedia mentions that the earliest modern naturist club has been created in British India in 1891 - this is more or less the time the Game of shadows takes place1. Since Mycroft is quite interested in novelties, it is quite possible that he has joined this club, even if only remotely (the exact opposite might be also true - before 1840, public nude swimming was socially acceptable in Great Britain, so Mycroft might be very old fashioned, but it seems too far fetched as Mycroft is not going for a swim).
1 - I don't think it has been exactly stated when the movie takes place (but for example the Mayerling Incident from January 1889 is mentioned), but naturism became quite popular in Europe early in the 1900s
Because he can
Do you care if you are walking naked in front of your dog? In his house, there is only his servant, his assistant and a woman. The first two are just a part of the furniture, the last one is...well, just barely a wife to a doctor. Who cares if those lowly creatures see him?