So I had always heard this is the case, and now believe this must be a common misconception. See Table 6 in this file from the Office of National Statistics:

http://www.ons.gov.uk/ons/publications/re-reference-tables.h...

While from 2004 to 2012 total UK fertility increased from 1.80 to 1.98, the fertility rate for non-UK born women actually /decreased/ from 2.50 to 2.29 (albeit in a slightly messy non-monotonic fashion) while the fertility rate for UK born mothers /increased/ from 1.69 to 1.90 quite monotonically.

It would be interesting to see the stats including 2nd and 3rd generation immigrant mothers (i.e. born in UK both with parents born elsewhere).