Here in the UK, I hear those as very different, although most people would still understand if they were swapped round. I do think many US Americans will hear the two as interchangeable.
(I'm assuming "… proceeding a next action…" here means "proceeding with an action." Is that right?)
I hear "hold on" as "stop what you're doing" just as yes, we say "hold on" to stop someone from proceeding a next action. "Hold on" can be an expression complete in itself.
I hear "hold up" as "interrupt or delay something else", as for instance "Hold up the traffic (until the procession has passed)". That works only with a stated object.
That the criminals pointing their guns at a bystander are carrying out a "hold up" is a co-incidence, isn't it? Similarly, this doesn't touch on "You hold up that end of the shelf while I fix the bracket" or "Grab something solid and hold on."