Tally up then
You need a fairly precise voltage meter, a fairly accurate timekeeping device, and a logger that records timestamps and voltages. Say, you record single precision floats with unix timestamps, that's eight bytes per measurement, times the number of measurements (say, once a second), plus some overhead. That setup times the number of places where you want to record, which might be half a dozen or less for the UK.
Then multiply the cost of the device by some factor to make up for government inefficiency. Let's say, ballpark, two thousand pounds per device, installed at convenient police stations, around the country, assuming they have mains power. Or perhaps utility offices. Or something like that. Then there's some small change for someone to drive around and visit each device at least once a year to see & certify if it still works properly, gather up the data and stuff it in an oversized database in the lab.
In short, peanuts, as these things go. If you want to record that stuff yourself, sure, go ahead. It'll cost you less, even.
Continuing, and in comment to various other comments, this sort of thing is _of course_ falsifiable, and so you can use it to frame someone else assuming the prosecutors aren't doing that already. But the thing isn't that it can be broken, but more that most criminals are stupid and the rest of the crime is more sort-of ``casual'' as in perpetrated on the spot out of desperation or other equally pressing circumstances. Most of the time investigators wade through the shambles of human endeavour gone wrong rather than the results of deliberate manipulation. Which is why when that happens and it's done professionally enough nobody is the wiser that manipulation happens. Which suits the perps, spies and such, just fine. Yes, you can be framed and no, if it's done well enough it's not going to be detected. But the likelyhood of that happening is for by far most people well below the likelyhood of dying at the hands of a suicide bomber, which in turn is lower than dying by traffic accident. So. It's not really a concern.
Of course, if the forensics take themselves seriously as scientists they must of course include the possibility of malicious input. But since it's well known in computing circles and still everyone forgets about it, I'm not so sure the forensics people manage to pay more than lip service. Right up until we discover that forensics are just as secretive for much the same reasons as bankers. But before that, there's simply not enough information to go on, so nobody will be raising enough hell to improve the situation.
Still, to me this is another tool and as such it isn't much of a privacy concern just like collecting dna samples and fingerprints at the scene of a crime isn't. Something has gone clearly wrong and it's good show if the hopefully competent and professionally impartial officials manage to find out what happened, who did it, why, and have them explain themselves in front of a professionally impartial judge. That's how it ought to work. Even the recording of power fluctuations isn't much of a problem like recording wood growth rings histories isn't a privacy invasion. It's useful data to have around much like historical weather data is useful to have around.
Contrast with preventively treating everyone as criminals by storing their fingerprints, dna, enough personalia to legally impersonate, and whatnot else ``just in case'', in what then turns out to be leaky databases, or just plain lose the data on the train for good measure. Juicy targets for identity thieves indeed. And then complain crime still hasn't vanished, and try and continue with battling thought crime. It's already happening, next to conditioning kids it's alright to be surveiled and biomeasured all the time. For school lunch, for checking out school library books, and so on.
As already has been noted, in a way this is quite an interesting development. I also recall DSP trickery to record monitor glow scattered from a wall and turn it back into a mostly readable picture. Since this audio thing depends on previously ignored ``noise'' and turns it back into meaning, well, what else is there? The CCD elements in cheap multi-megapixel cameras are notoriously noisy. Wonder what one can learn from the noise in digital pictures. And next to that, steganography may see a comeback of sorts.