Hey Peanutter,
Hospitals have some differences but are usually roughly similar
Most will want to do something 24 hours after your waters break but are sometimes happy to see how things go for 2-3 days, as long as everyone looks healthy and well (mum and bub)
The clock doesn't start on labour in general until "established labour" which is usually after 3cm dilated. Many midwives will delay finding out if you're "established" as long as possible to delay the ticking clock
Once you are in established labour the key for the timeline is that things are happening. So they like to see you "progressing" even slowly. If it is not happening then hospitals will generally suggest induction. Again they like to see you "progressing" and will usuallly monitor bubs after they go down this path. As long as you are "progressing" and mum and bubs is ok they'll leave you be for a few hours. They will keep turning up the syntocinon every 1/2 and hour for a few hours to see if they can get you into "established" labour. AFter about 4 hours most units would be starting to talk C-section if you still weren't having regular contractions. If you had any other form of induction like gels or ruptured membranes they would first try a syntocinon drip. (As above)
Pushing 2 hours seems to get people twitchy and talking about intervening, but once again, sometimes they don't find out you've been pushing for a while, depending on the experience of your midwife or if there is someone aware of your pushing.

This is a rough recollection of my experience both working on labour units and being a patient
I am not at all agreeing or disagreeing with the timeline, just recollecting it!
Hope that helps. And maybe others will have more info.... hope so. My memory is rather affected by baby rearing and sleep deprivation so am very happy to be corrected too