I’m afraid, unlike @davidallengreen, I don’t find @RobertCraig3’s analysis persuasive. Here’s why.
The key section here is section 45C of the Public Health (Control of Disease) Act 1984. (In fact all the relevant legislation here comes from 2008 amendments). 
Key points. The broad power (in (1)) is wide. (2)(b) tells us the power can be general in nature. (3)(c) tells it can impose restrictions in relation to persons. And (4)(d) tells us that such restrictions can include “special restrictions”.
What are “special restrictions”? See (6).
The drafter there picks up on (“incorporates by reference”) drafting from later sections (section 45G(2) being the relevant one for us): it’s the type of restriction a magistrate could impose (or rather a subset of those restrictions, some being excluded). These include (j):
With that background, a couple of claims made by Robert seem to me to be dubious.
First, he suggests that regulations can be made only in relation to people who are or may be infected. That seems to me to be wrong. (a) It ignores the “general nature” wording in s.45C(2)(b).
And (b) it reads into a qualification of the powers of the magistrate under s.45G (see 45G(1)) a qualification that is just not there in s45C (and is not imported merely because the drafting of s45C incorporates by reference some of the drafting of s45G).
Second, I see no basis for his contention that regulations can *generally* impose only more limited restrictions than a magistrate could impose (apart from the clearly indicated limitations in s45C(6)(b)) - an exception that rather shows the rule). Again, see s45C(1) and (2).
Third, I don’t agree that there’s a breach of s45F(6). That requires there to be an appeal against restrictions imposed under (“by virtue of”) regulations.
But the Regulations don’t make any provision for restrictions to *be imposed by virtue of* them. They just (directly) impose restrictions. So section 45F(6) doesn’t bite.
(Which is unsurprising: it would be odd, to put it mildly, to have a Minister’s decision to make general regulations imposing general restrictions appealable to a magistrate: judicial review in the High Court is what is called for.)
I therefore don’t agree that the legal basis (“vires”) of the Regulations is seriously questionable.
That said, David is right to draw attention to some distinctly vague and somewhat untidy drafting. Listen to podcasts.apple.com/gb/podcast/bet….
And Robert is also absolutely right to stress the urgent importance of restoring effective Parliamentary scrutiny, by virtual means, ASAP.
PS I also see some force in the argument by Tom Hickman QC (and others) here, namely that 45G(2)(j) (in the light of common law presumptions in favour of liberty) can’t be read as justifying qualified house arrest. coronavirus.blackstonechambers.com/coronavirus-an….
Though the argument that regulation 6 effectively imposes “quarantine” (expressly forbidden by s45C(6)(b) seems weak, given that there are numerous “let outs”, a point developed by Jeff King here: ukconstitutionallaw.org/2020/04/02/jef…
