Actually, it's real easy to beat the UK early while at war with Mexico. I do this every time to the UK.
1) Make some forts in NY, Maine, and the access points around the Great Lakes.
.....a) You'll need the forts to fight a defensive war, giving you the advantage.
.....b) Make sure you have leaders, because the UK's Canadian forces are typically leaderless.
2) Build up all your units as fast as you can prior to war (don't get irregulars, if possible).
3) Make sure you have at least one Army Tech advantage in one of the 5 categories, either better Morale, or Guns.
4) Make sure you have a naval presence. You'll need to do hit and run at first, but as time goes by, you'll need to keep building ships to clear the sea lanes, as the blockade will take effect. Don't be afraid to leave a battle, and go back to port to refit.
Then declare war on Mexico. Typically, I fight in Texas until I can get an edge, and then the Mexicans are on the run, and I can get 100% Warscore after a number of years.
In the meantime, the UK will come across the border. You need to fight a defensive war, and they will wear down fast, because you are fighting from forts on defense (i.e., force multiplier). Your leadership will carry the day too. Every little bit helps. Then, you'll be able to siege each and every province in Canada. Keep you small Navy alive ... and keep adding to it. Don't be afraid to retreat your fleet when you are outgunned (which happens often).
The UK will eventually ask out the war, but just be careful because they are the war leader. Make sure your demands from Mexico are settled. I've captured the entire US from Mexico in one war, while sticking it to the UK.
(And don't be afraid when you see their stacks of 75 points come over. The UK whittles away fast, and you can rotate armies too.)