EDITED TO ADD: Please also read the other answer by cbeleites unhappy with SX about the risk of heavy metal contamination. The information in my original answer (below) is still accurate regarding microbes, but you also need to worry about heavy metals, and a water filter will not remove those. Apparently the low pH of bog water makes the risk much higher than I realized. That makes the historical mining practices of the area extremely important to your decision about whether and how much of the water to drink.
Yes, a Sawyer water filter (or any other reputable brand of water filter) will remove microbial contaminants and make water safe to drink. The exception is if the water is chemically contaminated, eg by a nuclear plant, factory or mining. There is some history of mining in the North Pennines, so it would behoove you to look up where the abandoned mines are in advance, and try not to drink water from nearby. But as a general rule of thumb, if the nearby vegetation looks healthy, it should be find to drink filtered water from that site as a one-off. If you frequently hike the same route and regularly drink the water, then you might worry about longer-term health effects, and should do some more research about the historical land use along that route and possible chemical contaminants.
The issue with peatland water is that it tends to have lots of suspended vegetative particles that will quickly clog a filter. (This is based on my experience with the water in Dolly Sods, a peatland wilderness area in West Virginia, USA.) As the filter gets clogged, it will take more time and effort to push water through it. And Sawyer water filters are quite slow to begin with. Make sure you bring the cleaning syringe with you. You may need to flush the filter several times during the process of refilling a 2.5L water bladder. (And remember to only use already filtered water for flushing.)
I have found with the Sawyer water filters that the pouches are extremely difficult to fill from anything other than a water tap. If you try to fill the pouch by pushing it under water, the water pressure compresses it, so there's no empty space to fill with water. When you pull it out of the water, you have only half a cup or so in the pouch. It's much faster if you bring a spare water bottle, fill the bottle from your water source, and pour from the bottle into the pouch. Or if you have a bottle that the filter fits onto, you can attach the filter to the bottle and drink directly from it, using the filter like a straw. But then you'll have no way to flush the filter if it gets clogged, since you won't have any clean water. So it's probably best to test that method while you still have water left in your drinking bladder, to find out how much peatland water you can drink through a filter before it gets clogged.