Mathematically, this is based on the assumption of the infinite and the continuous.
However, quantum mechanics tells us that the world should be finite and discrete. Therefore, measuring the coastline of England naturally has nothing to do with limits. Of course, in practice, infinite measurement precision (not to mention the uncertainty principle) is impossible, so obtaining the "accurate" length of England's coastline—at least within the current framework of physics—is impossible, but that is another issue.
The meaning of "coastline" breaks down far before you reach quantum levels. I challenge you to go to the beach and pick a square inch where the "coastline" splits it half and half, this part England, that part the sea.
This is also ignoring the time component, where individual fluctuations in water will change the measurement, tides will as well on a daily and monthly cycle, and as years go by erosion and global climate change will shrink the land and grow the sea.