How would one go about building a similar distributed testing setup for end-to-end tests where a sequence of tests have to be run in particular order.

I reckon that would be solving the wrong problem. End-to-end tests should be independent of each other, and tests should never be dependent on the order in which they are run. End-to-end tests might be longer as a result, but managing the complexity of test dependencies will quickly cripple any system that uses this approach, I imagine.