I'm interested to find out what sort of medical problem is unforeseen so that you couldn't stock up in advance, not urgent/life-threatening enough that hospital isn't an appropriate solution, and yet urgent enough that waiting a couple of days means someone would be "screwed"??
There are medications that they are very unwilling to give you early or more than you need for the standard period of time (think any very strong painkillers for examples).
Then there are meds that aren't necessarily carried in large quantities or by every pharmacy so if your regular one doesn't have it on the day it's meant to be in you can run into problems.
I wouldn't necesarily say that you'd be "screwed", but you could potentially be in a lot of pain or require additional medications.
And then there are the unexpected things, like an acute infection, the emergency doctor might prescribe you something and you may need to start it ASAP but if there isn't a pharmacy open near you, you could be in trouble (some infections can get very nasty pretty fast, especially if you've got underlying health issues).
Getting the medication in those cases could mean the difference between a standard (strong) course of antibiotics, spending time in hospital, or a very long recovery at home, it can apparently take months to recover from some infections if they're not spotted and treated fast enough (look up on the effects of delirium which you can get from a bad water or chest infection).
Having said that, there is normally a pharmacy open somewhere relatively local in an emergency, and I suspect the duty Doctor on call, the local surgeries and I would seriously hope that at a pinch 111 should have the details as I believe the arrangements are formal and organised well in advance.