It depends very much what sort of redness/swelling/pain. An eye that is red inside, feels gritty and sore, and that I think is probably conjunctivitis, I'd try the pharmacist first, if simple home remedies like compresses, antihistamines, or eye drops from the chemist didn't help. Then if it still looked serious, optician/GP.
If it looked more swollen, and had pain in a way that I thought might mean an infection, something in my eye, or damage to the eyeball etc, and it appeared to be getting worse in any way, then either optometrist during the day or A&E at night. If it affected my vision, then definitely.
If it was more of a visual issue, like floaters and flashes, then depending on the local area, whatever it took to get seen at eye department - in my case, opticians first, who then referred to GP, who then referred to hospital , and that was actuallyl very inefficient and could have led to serious issues if I hadn't been insistent about getting the delayed referral phoned in as urgent. I don't think in this area you can just show up to the eye department, even if you have had previous issues (I was told that I should go in if it happened again in the other eye, which is very common, but it's not clear quite how I'd go about that now!).
But just red and sore, I'd make a judgement depending on previous experience about whether I felt OK waiting til morning or not, depending on what local facilities are. I wouldn't go in for just any redness or some swelling of eyelid, or I'd be in there all the time! But I know what my eyes are like, how allergies affect them, what is different, etc., so I'd make the judgement on that.
I wish optometrists/opticians offered an appointment that was clearly for eye health. Mine only seem to book sight tests, which I don't need again. Or I could probably ask to be seen, but it's not clear how much they would charge or what type of appointment to book, and if I had to go back for repeat visits, it's still not clear how/when I'd need to pay, but I'd feel guilty if they just said come back for checks. Mine isn't part of the minor eye conditions service, which would help.