This is now Alma Church.
According to the ever-useful churchdb:
It was founded in 1871 as a daughter church of Bethesda Chapel in Great George Street, Brandon Hill, and was called "Clifton Bethesda" to distinguish it.
I didn't remember noticing the original Bethesda Chapel on Great George Street and apparently I didn't miss it—it seems from churchdb that it was hit by a bomb during WWII and later demolished. The KYP map from after the war shows its outline on the corner of Charlotte Street South, tagged "Ruin", and there's a picture of it. It's back to being grassland on Brandon Hill now, pretty much exactly where this earlier photo was taken.
Bethesda Chapel was founded by George Müller, one of the founders of the Plymouth Brethren, and director of Ashley Down orphanage, looking after more than ten thousand orphans in his lifetime.
GPS Coordinates:
51.46232, -2.61312
Location: Clifton