Certainly the description sounds more like what we called Red Rover. (The person to attempt the crossing being singled out by the defending side with the chant "Red Rover, Red Rover, send <person> right over.") British Bulldog, as played at school and cubs, was IIRC, one entire side trying to get past the other, with just tackling and not the linked-hands thing.

"Primary school" isn't well defined. To me, "primary" would be "infants" (Years 1, 2 and 3 in modern parlance, I think) and I don't recall either game from there (but then no-one wanted me playing with them anyway). I do recall on at least one occasion it being banned (the ban must have lapsed, or been forgotten and a reminder was required) at juniors, due to the playground being tarmac. Infants was explicitly CofE (likewise cubs), juniors just woolly state.