The oldest AMS prize is the Bôcher Memorial Prize (1923). The MAA has the Chauvenet Prize (1925).
The London Mathematical Society's oldest prize is the De Morgan Medal (1884).The Royal Society's Copley Medal (est. 1736, first awarded 1746) is given to mathematicians and also to other scientists, as is their Royal Medal (1826). Their Sylvester Medal (1901) is only given to mathematicians.
The French Academy of Sciences awards the Prix Poncelet (1868). Their Leconte Prize (1889) https://en.wikipedia.org/wiki/Leconte_Prize is sometimes given for mathematics. (Note: these are Wikipedia links, good for initial research, but be sure to verify).