I'd ridden the A24 before the CSH changes and written a blog post about it here.
For the before-and-after comparison, I rode it again, now most of the blue lanes are down, just before the grand opening.
Riding North, the first problem I had was on the approach to Colliers Wood, with extremely heavy traffic but no blue lane. The blue lane starts after Colliers Wood tube station. In general, I think the blue lanes have improved things in that vehicles tend to stay out of them. However, they are mainly quite narrow, and there's not enough room to overtake a slower cyclist and stay in the blue lane. When I rode on a weekday morning, there wasn't much of a problem with cars parking in the lane. The Stockwell gyratory was a problem. I didn't see any particularly worrying incidents going into town.
Coming back, it was a different, more dangerous story. There was more congestion and motorists seemed to be encroaching more into what should be cyclists' roadspace. Take a look at this video clip (below). You'll see a bunch of cyclists approaching the lights. They're waiting to go straight on, and coasting to the red light but the car overtakes them into the blue lane, and then turns left across them. This is potentially dangerous.
My second clip (below) is taken at Clapham North. You'll notice cars changing lanes while cyclists attempt to filter through, and one in particular doesn't seem to have much concern for the fact that cycles are coming through on the lane he is entering (he gets shouted at). This is a classic piece of car-centric road design with no consideration given to cyclists. There are two fairly narrow lanes going into Clapham High Street, but the road narrows into a single lane. This is done to improve traffic flow, but it doesn't work very well because the traffic is backed up. There's no advantage to be gained for motorists filtering up the left-hand lane because they then have to merge with the right-hand lane a few metres further on. If the CSH designers at TfL were serious about making life safer for cyclists, they would have made the left-hand lane a mandatory cycle lane, given there's enough space. But that would go against the principles of maximizing traffic flow at all costs, which are given higher priority than cycle safety.
Based on what I saw, I think an inexperienced cyclist would not be tempted to ride this. There's too much intimidation by motorists, which in the main is not deliberate, just careless, inattentive or inconsiderate, but the crucial point is, the CSH does little or nothing to prevent it. The blue lanes are advisory rather than mandatory, and there is no physical separation from motor traffic. Bear in mind that many of the cycle lanes and all the bus lanes were there before the CSH, so all the CSH has done is make the presence of cyclists a bit more apparent.
I don't think the route has got any faster. The main problem is traffic lights, of which there an awful lot. It's sad to report, but those cyclists who choose to jump the lights enjoy a faster journey, more roadspace and less intimidation from motorists.
So in summary, I think the blue lanes have improved the situation for cyclists on this route, but not by much: not enough to attract new cyclists; not enough to reduce the number of accidents significantly, and certainly not enough to create a cycling 'revolution'. You would not want a child to cycle this route. They've done pretty much the bare minimum. There's very little in the way of new roadspace for cyclists, or safety measures, and nothing that prioritizes cycles at the expense of traffic flow. It's the same old story of the safety and convenience of cyclists coming last on the list of priorities.
This CSH cost around £10M. I think that £10M could have been spent making LCN#3, which is in the same direction as CSH#7, faster, safer and more cycle-friendly. That option would have given us a route that although longer by distance would have been faster, because of fewer traffic lights. It would have been much more pleasant, because it follows quiet roads. It would have been safer because of much lower traffic levels, and because of the ability to give better segregation between the cycle route and motor traffic. It would have been more attractive to inexperienced cyclists. In other words, Boris backed the wrong horse.