Assuming you have a quality filter which genuinely removes bacteria and protozoa, you are probably safer from viral infection than you think.
Viruses are quite likely to be present in upland water via contamination from human and animal faeces. But they don't tend to exist in isolation - they are normally attached to larger organisms such as bacteria.
I've discussed this at length with a filter expert who was involved in a large test for the UK military. They found that quality filters were removing viruses effectively, even though viruses were theoretically small enough to pass through the filter.
Based on this advice I've drunk some heavily contaminated water in the Alps using a good filter, with no ill effects. I also walk weekly on Dartmoor, where the water is quite infected by humans and stock, and the filter has protected me.
To be doubly sure, you could carry some purification tablets for use in sketchy situations, but I haven't felt the need and I've lived to tell the tale.
Personally I'll drink unfiltered if the source is fast-flowing and high on the hill, but otherwise I filter. In the UK the risks are modest and many people drink straight from the stream. But the consequences of an infection can be pretty unpleasant, and as filtering is so easy I prefer to be safe.
Most portable filters are not effective in removing chemicals. The three main issues are:
- Run-off from mining
- Run-off from agriculture
- Algal blooms.
In the UK, it's normally fairly trivial to avoid these contaminants.
In areas that have been mined you might want to ask locally if you are planning to camp low down. On the tops you should be above any mine-works.
Personally I avoid drinking water below arable farmland - this is not usually an issue in the hills if you plan ahead - just source your water on the higher passages of your walks. On coastal paths I'll carry enough water to get me to the next reliable source. With a filter, you can drink safely from the taps in toilets, from fountains, and from similar sources without having to resort to bottled water.
Blooms are rare in the UK and should be obvious if encountered. Given our climate, there is usually a running source above the infected body of water.
A quality filter should be all you need in UK conditions
So with a little common-sense and planning a good filter should see you through. There's no need to exaggerate the risks, but it's worth taking basic precautions whenever there is any doubt about water quality.