An umbrella alone probably won't cut it, either. Wind can be a big issue. Folks usually recommend a hooded rain jacket or raincoat. I have a soft waterproof hat with a brim that I can wear under the hood. Ever so stylish (not), but it keeps rain off my face even if I'm not using an umbrella.
I spent a lot of time last summer in Normandy, Brittany, southern England and Wales. I had much less rain than I expected, but I know I was lucky. Just about every day had a period when it looked as if it would rain, so I carried the rain jacket with me almost all the time. I didn't want to have to waste time later on, going back to my hotel for rain protection. I managed to cram the jacket into my purse. The umbrella accompanied me only on occasion.
I think waterproof shoes are a good idea. I might dare water-resistant if I already owned a good pair (not Scotchgarded tennis shoes--they wouldn't be supportive enough) and was planning to take a second pair of walking shoes. In that case it wouldn't be a disaster if the shoes didn't dry overnight. But I'm sure Scotland trends wetter than southern England. (Scotland/Scotchgard--never noticed that before.)