Go to Antique Guns at the related link and describe your firearm in as much detail as possible (and in the correct category or a couple of them will give you a hard time). The people there will give you an estimate of the value and help you determine if you really need a professional appraisal, and if so, guide you in the right direction.
There are several specialized auctions houses in the U.K. (and the U.S.) and other general auctioneers like Christie's and Sotheby's who have specialized arms auctions from time to time. They will freely advise you as to what you might expect if you put your item up for sale. Fees can be high, however. You might also try one of the U.K.'s best antiques arms dealers, Peter Finer, who I can wholly recommend as being fair in his dealings. However, he only buys and stocks the finest examples.
Perhaps a good place to start would be to find a copy of Flayderman's Guide to Antique American Firearms- you may be able to request a copy through your local library. This is a fairly comprehensive book with good descriptions and illustrations of thousands of antique firearms. You might also check for gun dealers in your area that know antique firearms, or check out a local gun show, where dealers in antique firearms have been neatly gathered in one spot for your convenience.
In firearms, antiques are LEGALLY defined as made prior to 1899, OR muzzleloading firearms. Newer than 1899, they are old, but not antiques. My best reference is Flayderman's Guide to Antique Firearms. You may also find a dealer or collector that can share some information with you. Check for gun shows in your area.
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