The traditional Andurien salute to the Captains-General is one finger. Guess which one. ;D
Seriously, I'd think that House Davion would do the British-style salute, given that a large part of their cultural identity comes from that region.
For Kurita? I'd think a bow for formal occasions, but in a military setting, they'd use salutes. Both the IJN and the JASDF, if I'm not mistaken, use military salutes except in formal occasions.
The one I don't know about is the FWLM, especially pre-reform. Since they had a multiplicity of different uniforms, maybe they had a variety of different ways of (and attitudes on) saluting depending on the particular province? I could see the Marik Commonwealth, for instance using a salute used in the Balkans/Hungary/Romania/Bulgaria/central-south-eastern Europe, for instance, and Regulus using a Commonwealth-style salute (as per their heavy Indian character)
The Marians don't do the "Ave Cesar!" thing made so unpopular by that 20th century political party I'm avoiding mentioning to steer clear of Rule 4?
While that would definitely be something the fanboy rulers of the Hegemony would do, I'm sure that the real-world devs didn't need the headache of having to say "no, they're emulating a different ruthless empire, stop shunning us!" every time that would be brought up...