I have, what I believe to be a 1797 BRONZE cartwheel 2 pence. However, when I look online for values, everything seems to be COPPER. When (rarely) bronze is mentioned, there are no values given. Can anyone help?
If I understand correctly, they're all copper. The reference you've seen may refer to proofs, some of which were bronzed copper. Other proofs were gilt copper.
Cartwheel Two Pence and One Penny are pure copper. Easiest way to tell apart is the the Two weighs 2 ounces and the One weighs 1 ounce. Tony Clayton, who is a member here has the most upto date valuation site. http://www.coins-of-the-uk.co.uk/values/index.html#index