This page gives you information on what type of flag you can fly from your church.
The Earl Marshal gave directions in 1938 (which are still applicable today) that the flag proper to be flown upon any church within the Provinces of Canterbury and York is "The Cross of St George and in the first quarter the escutcheon of the Arms of the See in which the church is ecclesiastically situated".
In simple words, this means the flag of St George with the diocesan arms in the top corner nearest to the mast.
More detailed advice has been given by the Garter Principal King of Arms:
1. The Arms of the diocese are to be placed in the first quarter only (see above).
2. The shield of the Arms should fill the space and should not be ensigned with a mitre.
3. New flags should be made to an approximate ratio of 5 x 3 which is larger than, or more square than, ordinary flags to enable the Arms to be included. Shields may be added to existing flags of St George without altering their shape.
4. The shield of Arms should be of the same material as the flag and inserted. The larger charges on the shield are also to be inserted but smaller charges and shading can be applied in oil paint. All heraldry is read from the flagstaff and charges on the shield must face the same way in relation to the staff on both sides of the flag.
5. It is not compulsory for the flag to be flown.
There is a guidance note on flags and military colours here.
Churches may also, if they so wish, fly the Union Flag on the same days when it is flown from Government and other buildings. Details can be found here.