Well, I hope we've got a back-up because the internet is open to everybody, friend and enemy alike.
Of course there's a fallback. It's just horrendously inefficient and would be unable to deliver the level of care that we offer today.
There are things however that absolutely need the computers - imaging is one. X-Ray film simply hasn't been used for many years, and the more advanced imaging (CT/MRI) requires computers to generate reconstructions.