I looked a bit closer at the global usage and I think the truth is that usage varies greatly. Here are some stats for all the ways with bridge=yes set:

Global usage:

oneway=alternating + priority=(unset) -> 1944
oneway=alternating + priority=backward/forward -> 380
oneway=(unset) + priority=backward/forward -> 1037

New Zealand usage:

oneway=alternating + priority=(unset) -> 1176
oneway=alternating + priority=backward/forward -> 347
oneway=(unset) + priority=backward/forward -> 26

Given that at least two variations appear in wide-spread use and usage varies regionally, I think now would be a good time think about how we want these tags to be used. I tend to favor priority= without oneway= simply because it is easier to understand (for an English speaker) and would hopefully lead to less mistagging in the future.

I also wonder how a one-lane bridge with a traffic light would be tagged (oneway=alternating + priority=(unset)?).

If you don't mind, I'll try to see if this ambiguity was already discussed anywhere and if not ask the community. Given that I'm traveling, it'll likely take me another two weeks to find time for this. For the time being I'll refrain from changing any existing oneway=alternating tags.