We were on the Isle of Skye in Scotland in August 2014, where it absolutely poured. Light, ankle-high, water-resistant trekking shoes worked fine, although we stayed inside when it was raining the heaviest, and only ventured out when it wasn't coming down in sheets of water but the ground was still quite wet.
For our trek across northern England this June (Hadrian's Wall Path National Trail) we had different, all-leather waterproof shoes (didn't cover the ankles) and regular Gore-Tex hiking boots, like we use when summiting Colorado mountains. One morning, half an hour into the day's trekking stage, it poured and poured. The waterproof hiking boots weren't and I spent an hour in a pub, in front of the fireplace, drying out. I brought along gaiters which are designed to be ankle covers, protecting your socks and lower legs if you're wearing shorts, or keeping things from getting up your pant cuffs if you're wearing long pants, and those can offer water protection, but I didn't bother digging them out of my pack, as the rain was so sudden I would've gotten even wetter in the time it took to put on the gaiters.
Gaiters might work for you, but a pair of Wellies really seems like a lot to haul around, unless you're wearing them every minute of your trip. A pair of good hiking boots, waterproofed, and maybe a $20 pair of gaiters as additional gear, should work. The rain in British Isles has a way of exceeding even the best-prepared walker, but Wellies seem like overkill.