Well, let's see...
The RAF:
Lancaster I - (2) .303 nose turret, (2) .303 upper turret, (4) .303 rear turret
Halifax Mk.III - (1) .303 nose, (4) .303 dorsal turret, (4) tail turret
Stirling I - (2) nose turret, (2) dorsal turret, (4) tail turret
Lincoln I - (2) .50 nose turret, (2) .50 OR (2) 20mm dorsal turret, (2) .50 tail turret
The U.S.:
B-18A - (1) .30 nose, (1) .30 dorsal, (1) .30 ventral
B-17F(early) - (1) .50 nose, (2) .50 "cheek", (2) .50 dorsal turret, (1) .50 radio skylight, (2) .50 waist, (2) .50 ball, (2) .50 tail
B-24D - (3) .50 nose, (2) .50 dorsal turret, (1) .50 tunnel, (2) .50 waist, (2) .50 tail
B-25C/D - (1) .30 nose, (2) .50 dorsal turret, (2) .50 ventral turret, (2) waist (optional), (1) .50 tail
B-26B - (1) .50 nose, (2) .50 dorsal turret, (2) .50 waist, (2) .50 tail
In comparing 4 of the RAF's bombers against 5 of the U.S., I am having difficulty in seeing how the RAF was "on a par" with the U.S. heavy bombers. So I looked at the U.S. medium bombers and lo and behold...the medium bombers were comparable. (Except for the B-18, I just tossed that in for the heck of it!)
Bottom line, the U.S. heavy bombers were heavier armed, both in caliber and quantity, than their RAF counterparts.
With the exception of the Lincoln, the RAF bombers were woefully under-gunned, no matter how you look at it.