Needs more information. The way this militant liberal movement (pretty ASB in itself) comes to power (even more ASB) would need to be defined to give us an indication of how they are likely to treat with Napoleon. You're going to get one of two possibilities really: the British "liberals" looking up to Napoleon and essentially acting as his backup, a la how the UK supports the USA in the present day, or they're going to see themselves as more important or equals of Napoleon, causing the two allies to bicker over how to divide their winnings, probably causing the fragmenting of the alliance - France and the UK had overlapping ambitions so it's going to be hard to find a compromise.
Also, it's important to remember that Napoleon wasn't a liberal. He was a visionary who manipulated a popular movement to come to power. He declared himself Emperor in 1804 remember, that's not really a very liberal act. If he does that in this timeline, it's going to be no holds barred and the two countries will go at it like it was the mediaeval era all over again - and this time it will be the British who are the less acceptable to Europe when it comes to alliances.
But really the whole scenario is ASB from the start - Napoleon had a lot of popularity among the masses in Britain, but he would never get a friendly Parliament. He was anathema to British politics.