How about the Suez crisis goes differently? The UK regains the canal (and maybe occupies a larger buffer zone to protect it against Nasser trying a comeback), but Egypt is still covertly supporting insurgents in the area. It's still much smaller scale than Vietnam, but then, the UK is a much smaller power by this point. Plus, success in Egypt might convince the Brits to try to hold on to more of their colonies elsewhere.
Other possibilities:
-They decide to pull a France and intervene more frequently in their former colonies, eventually landing in a nasty conflict.
-This requires an earlier POD, but make the Brits try and hold on to India longer. That's a situation that's bound to get messy. For extra credit, combine it with the above (although British intervention in an Indian civil war is such an obviously bad idea that even British politicians could probably recognize it as a mistake).
-Others have mentioned the Mau Mau rebellion, but somehow make it go even worse.
-Likewise the Malayan Emergency; make the British do something sufficiently alienating that the Communists get sufficient local support to make it more deadly.
-Have the Brits try to prop up some neighboring colonial regime.
The big problems are:
1) Especially after Suez, the British have little remaining appetite for holding on to their colonies, so they are likely to pull out if a conflict starts going seriously south.
2) Any conflict big enough to resemble Vietnam is likely too big for the British to handle.