'Max',
I have no direct experience with your exact situation, but from personal experience with import from other countries I believe you have basically two tasks: (1) satisfying the US govt import requirements (which can involve three agencies; the US Fish & Wildlife Service, the Dept of Agriculture, and US Customs) and (2) satisfying the UK authorities export requirements (if any). With those documents in hand, then it's a matter of selecting an air-freight forwarder to move them out of the UK, and then selecting a freight clearing agent to expedite the USA import after arrival at the port of entry.
I would start by asking the USF&WS in writing what they will require in your situation, and since you have no "raw" trophies, and no CITES listed animals (I'm not sure about the Muntjac) their requirements might be fairly straightforward. Being "finished" and not "raw" trophies, the US Dept of Agriculture requirements might also be fairly simple. If they require UK export documentation, then you'll have to obtain that. It is ALWAYS a good idea to keep all documents associated with the trophies in case authorities at either end require further supporting documents (hunting licenses, receipts, etc.).
I would get started on this as soon as possible, as with any govt agency the wheels turn slowly.
I would not be surprised if there is a USF&WS office, or agent in the UK, and you might start by communicating with them. They might be quicker to respond that USF&W HQ.