I think you'd be wasting NHS resource with what is an overuse or overtraining injury. You say you did what was a long walk for you -- are you usually sedentary? It sounds simply as though you've used your body in ways it's unaccustomed to, and this is muscle soreness.
The other thing it might be is plantar fasciitis, which is very painful, but there's not much you can do, and it's certainly not worth a visit to an injury or emergency clinic!
You need to learn to work through your foot carefully when you walk, go barefoot as much as possible, and look online at some reputable physiotherapy videos for simple exercises to strengthen and stretch your ankles and feet. There are very simple exercises you can do once the immediate pain has worn off.
Also check your shoes: are they supportive enough, do they have a little bit of a heel, and arch support? One of the reasons I utterly detest the "fashion" for those awful Ugg bots is that they are so bad for people's feet. But so also are cheap plastic shoes, and those flat ballerina shoes.
I find that generally, people really don't know enough about self care of their own bodies: do you do something like yoga or Pilates, which will teach you about correct alignment and posture. Do you stand and walk and run tall and spine aligned, and with your core engaged? Do you think about the way your skeleton aligns? These things all have an impact on your gait, and on the health, strength and fitness of your feet.
IME, you'll hobble today, but if you walk "mindfully" and think about your alignment and the way you roll through your feet from heel to toe as you walk, the muscles will relax. When you're not active, try the RICE approach:
Rest
Ice
Compress
Elevate
I'd use the Ice and Elevate aspects: fill a 500ml water bottle not quite to the top, freeze it, then when you're sitting (with good alignment not slumping) use the frozen water bottle as a roller for your afflicted foot. It will hurt, but after about 10 mins you'll feel the benefit. For the office, I find a £2 wooden foot roller from the Body Shop helps enormously. Shoes off, of course!