Highly likely in almost any season and the footpaths may not be what you'd think of as footpaths.
I'm now 77 and I remember walking in the Lake District in the early fall of 1977. Even with good rain gear and serious Vasque hiking boots, I got drenched and had to put those boots in the drying room of my hostel. Fortunately, it was well equipped for drying out our stuff.
These days, I wear mid height boots (like these KEENs) that are waterproof and support my ankles. They've kept me from falling on rough terrain in and outside of cities many times. I wear them with over the calf compression socks (like these Sockwells) that keep my feet warm and dry no matter how many puddles or how much wet terrain I walk through.
You might want to reconsider your footwear. Last time I wore sneakers in Europe it was in Italy and it was a big mistake there -- not due to wetness but due to rough terrain in both the cities and countryside where we went.
See the info below and thoroughly explore it beyond the specific link I provided.
This link from Yorkshire Dales National Park gives advice and from it you can get to other parts of their website about walking in the area: https://www.yorkshiredales.org.uk/plan-your-visit/essential-information/take-care/
Please note this quote from it:
Clothing
Comfortable clothing/layers
Waterproof/windproof outer jacket
Boots with good sole pattern – trainers aren’t suitable except on the simplest of walks as they can slip and don’t support the ankle.
Hat and gloves
Spare warm clothing
Remember, when you’re resting, you may need more layers than when you’re on the move. Several thin layers give good insulation and allow you to adjust for the different activities dictated by terrain and weather conditions. Bear in mind that as you go higher it gets colder and windier.
Good luck with planning for your walk and with doing it.