French 51st Regiment/Infantry coat button from the period of the Napoleonic Wars button (1803-1815).
IP, I'm not as certain but then again I don't have the books that would answer it for sure. The design pattern of this button which a two digit number surrounded by a circlet with tufted/flanged endings and a dot or pellet and its zenith in a die cast two piece copper alloy with a convex front/concave back and a birdcage style shank does date to the period of the napoleonic wars as I said above. However, very similar patterns also show up during the Crimean War period which would be more in tune with your assessment. Further research into maybe a makers mark on the concave back coupled with definitive evidence that tunic/coat buttons during the napoleonic war period could had had a convex front/concave back button rather than what I have seen so far as just flat buttons with similar circlet numbered fronts might nail it down to be precise. The plain style circlet on most of the dubbed "napoleonic wars" buttons I have seen is not as prevalent in the later period, again supporting your assessment. Regardless, the design pattern (and pretty damn close to exact pattern) dates to an earlier period whether this button was made then or not.