OK - I'm an 'experienced' canoeist/kayaker, so what I regard as reasonable, is dangerous for a neophyte.
The EA is a public body, so they have to issue warnings to try accommodate the fact that people are bloody numpties.
The Thames is variable - are you talking tidal or non-tidal? If tidal stretch, then don't go on it unless you are experienced. Mostly because you are mixing it with powered boats, some big. Apart being a real pain in the arse to those boats if you get in their way, they are very restricted in their ability to maneuver so have right of way mostly (something a lot of canoeists fail to appreciate). Get in wrong and you will be squashed. Also the tidal current can sweep you quite a long way away pretty quickly, unless you are fit you might not be able to paddle against it.
Canoes are obviously an 'unpowered craft'. That doesn't mean slow; I could sustain 10kph for hours when I was fit in my K1, but most people can only manage walking pace and that for only 10-30min. If the river is flowing faster than walking speed (which is only 'slow flow' on most rivers), then it follows that most people will be swept downstream.
Basically, if you have to ask if a river is safe to paddle in, then you shouldn't be taking your canoe out on the river. It means you don't have enough experience to be out without someone to guide you. Stick to canals, small lakes or rivers with little or no flow. That might sound brutal, but it is honest. People drown when it goes wrong.
If there is a guide and some experienced people along, then it is a different matter. I've taken complete beginners down 'class 2-3' rapids in an open canoe and they felt safe with me (I'm a 'cautious' paddler). Class 2-3 would be the EA 'strong stream'.
The guide should know the waterway and the 'experienced people' be skilled at handling watercraft, able to do some teaching and help with handling multi-person craft and perform rescues.
