Well, looking at raw stats, there doesn't seem to be a correlation:
http://www.parliament...efings/snsg-04334.pdf
Page 2 in the above document shows a steady climb in the UK prison population, with significant jump in the period immediately following WWII, and then again in the mid 90s.
These jumps don't marry up to changes in immigration rates. Also, the figures aren't split by nationality, nor do are they subdivided by type (violent/non-violent) offence, so they're just a broad guide.