Not sure if this is helpful but I ride around with my little doggy in a burley bee with her lead (which is attached to a harness) tied at two points above the seats. When she's alone in the trailer, she stretches across both seats, when my daughter's in the trailer too, she takes one seat. I ride in peak hour traffic everyday (though not in London) and whether it's safe depends on how well you ride and how safely drivers drive. I am very familiar with my area and there are quite a few roads I avoid riding on because of the traffic or some other quirk that I think makes them dangerous.
You will always get very dangerous drivers on the road and sadly, that's the risk you take.
Pretty much all of the bike trailers/carriers have their own pros and cons but if they comply with all EU standards etc, then they are "safe" in the sense that they are unlikely to fall apart while riding.
I'm a very safe (some would say boring) rider but at the moment I'm not a good enough rider to ride confidently in traffic in central London. Whenever I go to London and see the tourists riding those hire bikes past Buck Palace (one of the most dangerous roads for bikes I've ever seen) I just can't watch....