Something like a breathable rain jacket for general wear. These are typically light enough to be worn all the time and if you get something loose fitting you can put a layer or two under them. The ones sold in shops with hiking gear are a good choice and they often have plenty of zipped pockets ! Some of them also come with a detachable inner flannel lining, almost like a second jacket, that attaches to internal zips. Can be very nice in colder weather.
Can be quite windy and wind chill can be a shock if you're not used to it, so consider gloves. A woolen scarf is a useful extra in a backpack.
Water-resistant foot wear if possible, but many soft shoes now are already "good enough" unless you're hiking or in seriously stormy weather.
You might want gloves. There can be quite a "bite" in the wind (wind chill) and you might find it a shock to the system given your home.
Avoid umbrellas. They're a nuisance to other people and won't stop you getting wet, while also being a nuisance to carry. The wind can also make them an absolute pain to use. More trouble than they're worth. A good jacket will do much better.
You can get breathable tops which are a good choice for walk around. I've generally found jeans fine for walking around Ireland (similar to the UK) all year, unless you're doing serious hiking. They'll manage OK with rain, but may get a little wet. Cheap jeans are also things you don't need to worry about getting dirty or damaged.
Make sure your backpack or over-the-shoulder bag is a good water resistant one and/or make sure you have some plastic bags to wrap the contents in. I typically fold up a couple of plastic carry bags and a couple of large zip-lock bags inside my own bag - very handy.
There is no such thing as "enough" zipped pockets. I speak as a person who liked to do street photography.