Just to explain a little bit more about OOH care. Overnight care/emergency vets etc is all pretty much the same thing.
Years ago, vet practices would all do their own 'OOH', the expectation from owners was lower, case load often lower, ability to do things was lower - so it was manageable. You generally ended up with a vet working the day, on call at night, hopefully not many call outs, and working the next day. Not many patients would be hospitalised overnight, or they would get a couple of overnight checks.
Nowadays expectations, demands and needs are much higher, and a lot of practices can't cope with covering night as well as days.
Some practices will have an OOH provider that they contract do (might be 'vetsnow' or something similar), this will be a dedicated emergency and OOH clinic. If you had a emergency at night (or sometimes weekends and BHs as well), you would do there, rather than your regular practice. Any in patients would often be transported there as well (and often then back to their day practice in the morning). This transporting around may be arranged by the owner, or the practice.
Some practices will have their own OOH. This can be done in lots of different ways - and don't assume they are all the same. Some practices still run the 'old way', with a vet on call, and working the previous and/or next day, therefore something is likely to end up lacking if people are working 36 hours or so straight. In patient care can be very variable - some places animals left on their own overnight, some on their own with a vet or nurse popping in during the night to check them, some just having a nurse, or trainee nurse in overnight.
Some places do have a true 'hospital' and a vet and nurse working out of hours - even this varies though. Some, such as one I worked at before had several branches, but the smaller branches were about 30-45 mins drive from the hospital branch - so a long way for a middle of the night emergency.
In all cases there will be an increase in cost for OOH care - consults out of hours are expensive - having a vet and nurse on site, or even on call, is expensive.
I can see reasons for all the above ways practices work - and some of the ones that don't seem ideal can work very well, and give excellent care, sometimes hospitals sound great, but can fall short, and everything in between. Don't let the OOH arrangements put you off a great practice, and don't join a practice just based on their OOH arrangements.