Britain didn't have slavery†. The 1833 act abolishes slavery in the Colonies, not in Britain itself. The existence of slavery in the colonies is why there were slaves in America to seek freedom by fighting for the British there.
So like I said, the British freed those on their side, while the colonists unsurprisingly did not do likewise. This is a recorded motivation for the thing we're talking about, fighting for the British in the revolutionary war.
†Britain didn't have chattel slavery, which is what we're talking about here, but it did have serfdom - which today we would consider a type of slavery - until the late middle ages or so when it ceased to make economic sense.
The Colonies weren't Britain. Their people weren't (on the whole) British citizens, if you knew a bit more history that year might stand out more. 1833 is just after Great Reform, the country has become more small-D democratic, expanding the franchise (the right to vote) to more men and eliminating "Rotten boroughs" which were often in practice controlled by rich people, including slaveowner colonists abroad.