The era of stylized or caricature birds and animals on English shotguns (Scotland was different) begins in earnest with William Palmer (1737 1812). (A caricature is a portrait that exaggerates or distorts the essence of a person or thing to create an easily identifiable visual likeness.)

You will see earlier examples certainly (Robert Rowlands guns, circa 1718) but broadly, grossly, generally, the trend begins with Palmer. Because he was one of the earliest engravers who consistently specialized in guns (for the likes of the Mantons) he was able to develop a method which produced recognizable images with just a few strokes of the graver. Prior to this guns were engraved by men from other metalworking trades such as jewelry. The end came with the emergence of exaggerated realism at the turn of the twentieth century in the hands of Harry Kell and his contemporaries. Nave engraving, as it is commonly called, spanned approximately the Victorian era.

What I learnt while researching British Gun Engraving was that nave engraving had several components. First the artisans were urban people who rarely saw game or indeed shot the guns they engraved. Second price was a serious consideration. Typically a gunmaker would request two penneth of engraving or sixpence worth if the gun was really special. Certainly a style emerged, or at least a consensus of what was appropriate, but this was more by default than design. Its a style that ran concurrently with others so its not much use when trying to date shotguns.

I hope this helps. With gratitude to Hugh Lomas.