Most dry-cleaning companies will do a decent job. Just remember to remove any enamelled badges (pips, collar dogs, etc) before you take it in, the solvents will take the enamel off. (from bitter experience )
Having had to resurect my mess kit after a dip in an ornamental pond (another story..), I can heartily recomend Blossom, Brown & Sycamore off Clapham Common. They removed all the pond weed etc & still only charged me their standard 3 piece suit rate (Â£15 a couple of years back). They hold a royal warrant or two, & do a lot of theatrical costumes, so are quite happy with Mess Kits etc.
Would go back without hesitation!