Hello London visitors! Welcome to the UK capital city. It’s great for cycling here, and we’re not just saying that!
VisitLondon.com (the “official visitor guide”) tell us that:
“The banks of London’s river Thames offer long stretches of traffic-free cycling. Most of London’s Thames-side cycle route is on the Thames Path National Trail.”
Sounds great! So let’s follow that link to the Thames Path website and check out their “planning a trip” page:
“The Thames Path is a wonderful place to walk, but PLEASE NOTE it is NOT a long distance cycle route. See our FAQs about cycling to find the short sections which can be cycled.”
Oh.
The Alternative Department for Transport is written by, and the personal opinion of, 
The Alternative Department for Transport 
eeerm, most of that Thames Path side is devoted to the towpath above Hampton Court, which is just on the edge of Greater London. Within London I have cycled quite a lot of the path — the Thames Barrier to Tower Bridge, most of which is also signposted as national cycle route 4, and upstream as far as Westminster Bridge with care for others. So within much of the capital city, yes, it is a good route for cycling. I commute to and from work along cycle route 4 between Deptford and the Millennium Bridge, even — depending on the mood of the wardens — past City Hall.
I meant ‘Thames Path site’, of course.