How do I clean my uniform?

[Polyethyl]

My uniform got drenched yesterday. It is a wool jacket. It's old, probably WW2.
It got splashed with muddy puddle water and it got polish (from my belt) on the waist.
I asked my colleagues and they suggested buy a steamer. Does a steamer remove dirt?
Should I dry clean it and if yes does that mean I should remove all the metal buttons? Then sew them all back on again afterwards?
I can't remove the brass belt hooks from the jacket.

This says they can be dry cleaned.
You shouldn’t need to remove the buttons, we don’t have to when we get normal stuff dry cleaned, but do ask if they thing there will be any issue and show them the polish mark, they may be able to treat it/give it extra attention.

https://www.wwiiimpressions.com/pages/care#:~:text=Garments%20made%20entirely%20from%20wool,contact%20water%20they%20WILL%20shrink.