I watched the video finally (apart from the last five minutes) -- mainly because I was wondering what this "constants stock market" thing was about.
It's an unfortunate argument, because he's presenting an unfair caricature of how science works. Leaving that aside briefly, though, the most serious flaw is that he's attributing perfection to experimental tests, and that's frankly unrealistic. No experiment is ever perfect, and as a result of you would of course expect that measuring fundamental constants might give different answers, either because of changes in technique, or different groups, or any number of other factors. Hence all the averaging, because it's sensible to try and remove and correct for these experimental errors. It sounded to me rather like Sheldrake was dismissing, or even mocking, this approach -- but it has to be the first starting point, rather than something brushed to one side.
As I typed that it occurred to me that this plays in to one of the dogmas he was talking about, the idea that the laws of nature are the same everywhere (or if he didn't say this explicitly then it must certainly be implicit). There is a sensible reason for this dogma, though. Well, two: firstly, it does actually fit with observations very well, and secondly, abandoning this dogma would essentially destroy any chance of replicating somebody else's results or testing them. If there is no requirement for two experiments, or two observations, to ever be the same, then the first one cannot be falsified. It would ironically hold back progress rather than advance it: Joe Bloggs could claim to have discovered some wonderful new pattern in, say, varying speeds of light, and then dismiss any attempts to test this that came out negative by explaining that the variation only happened that Tuesday in his backyard.
The bit about "intellectual phase locking" that was laughed at so derisively, almost as if it were an excuse, is odd too. It's a real problem. Call it fudging, call it something slightly more posh-sounding that means the same thing, it amounts to the same: scientists can have a tendency to "follow the crowd". They try not to, and over the decades increasingly sophisticated techniques to avoid this can be employed, but it does happen that one group gets hold of a result, and a bunch of other scientists get a little too excited about it. The most recent example I can think of would be the "faster-than-light neutrino" measurement of 2011(ish). An experimental group at CERN claimed that neutrinos travelled faster than light, which is supposed to be impossible. Dozens of papers, or even a couple of hundred, came out over the next few months, proposing various theoretical explanations of this. Then the first measurement turned out to be flawed after all. Excitement over. It seems remarkable that Sheldrake's explanation would presumably be that neutrinos did briefly travel faster than light, rather than that the experimentalists messed up!
Or you can have "fudging" when one group makes an error and nobody notices it for a while, so that the original result is allowed to survive. I speak from painful experience here -- long story short, these things do go noticed eventually, and corrected, but my impression again is that Sheldrake would claim that the first error was right after all!
In a nutshell, my criticism of his passage on the constants is that he seems to be attributing to scientists simultaneously the ability of perfect measurement but of hopeless vision. Whatever your views on the second part, whether you find science "disappointing" or not, the first bit is extraordinary. No measurement is perfect, or no measurement can be interpreted perfectly.