This BSA survey is helpfully repeated each year, so I've produced the above graph showing the trend.
Also plotted are the results from the British Election Study (BES) for 1987, 2001 and 2014.
Finally, there is one point from the Faith Debates / YouGov survey of 2013.
The sample sizes for the latest surveys were:
BSA - 2,878
BES - 20,881
YouGov - 4437
Each survey asked the question: Do you regard yourself as belonging to any particular religion? One of the answer options in each survey is the single category Church of England / Anglican. (BES gives Church of England / Anglican / Episcopal, to be precise.) The range of possible answers vary, and also the structure of the question (one or two parts).
None of these slight differences point to any obvious reason for the large discrepancy between the latest surveys. The larger sample sizes of BES and YouGov, and their agreement, may weigh in their favour, but it's hard to trust the BES result that there has been negligible decline since 2001.