I can't speak to the Christians, but even the most orthodox Jews I know support emergency services on the Sabbath -- and they'd absolutely drive someone to the emergency room if it came to that. Doing your best to observe a day of rest doesn't mean you can't choose to do work if it serves a greater good. (Whereas I'm pretty sure people won't die if they have to wait an extra day to order their camera equipment.)
And it's definitely complicated. Just because work necessary to preserve human life is allowed doesn't mean anything done in a hospital is therefore permitted -- a sizeable fraction of the work in a hospital is about bookkeeping, billing, insurance, scheduling followups, paperwork, etc which can wait a day or two without anyone dying.
In NYC, most hospitals have a special elevator that, on the Sabbath, continually stops on every floor. This is to help orthodox Jews comply with their commitment to not turning electric devices on or off--they just board the elevator at the floor they want and get off when it arrives at their destination, without ever pushing any buttons.
