Round works virtually anywhere around would work. The reverse is not true, as round has a number of definitions it doesn’t share with around. For example, it wouldn’t work to say that the edge of a circle is around, and I wouldn’t invite you to play an around of golf. But even though round works as a breezier alternative to around, round tends to create a casual tone, so around is usually safer in serious or formal writing.
British writers in particular are wont to use round in place of around. This substitution does occur in American English, but much less often.
Here are a few instances where the shorter round is used where around would also work:
He does admit there was the occasional cross word when his children were young and running round the house. [BBC News]
Early in the second half a man ran towards Lennon and attempted to punch and grab him round the neck. [Guardian]
The defendant then reached 60 miles per hour on a road with a 30 mile per hour limit and went round another roundabout in the wrong direction. [Birmingham Mail]