From what I understand, the Brown Bess was in service until 1838. The next service weapon I found was the Pattern 1853 Enfield Rifle. But what was used in between? The Pattern 1851 Minié rifle? That appeared in 51. I read something about a Pattern 1839 Musket, but couldn't find much about it. So what did the soldiers use between the Brown Bess and Enfield?
The Brunswick rifle was in service as early as 1836, and was actually still refurbished/produced until the mid 1880's! I'm pretty sure it wasn't the universal longarm, but definitely saw widespread use. It fired a .76 caliber ball, had two groove rifling.
The Brown Bess "New Land Service" pattern was used until 1839. In 1838/1839 the British introduced the new-pattern percussion musket. It was essentially identical to the earlier flintlock ones, except for the ignition system.
The above rifles still had flintlock examples serving with them, until everybody adopted the Minie'-pattern in 1851, in 1853 the Pattern 1853 Enfield Rifled musket came into service.
In the early '60s, the Enfields were converted to Enfield-Snider pattern breechloaders.
Edit for subtext, I think you mean the brownbess was in service until 1838