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I can find no evidence that this is illegal or widely considered disreputable. Acas explicitly describes a practice that is almost identical:

Working a 'week in hand': This means a worker receives their week's wage the week after it was earned. For example, if a worker begins a week's work on 1st January and is paid weekly, they would get paid for that week on or before the 14th January not the 7th January. If you use this system then when a worker leaves, they will be entitled to the week in arrears in their final pay.

I've never personally encountered this, and it may indeed be rare in salaried employment, but the fact that a pseudo-government source simply describes the practice in matter-of-fact terms in its advice for employers, without any hint that it's a thing they shouldn't do, suggests to me that it's not a red flag. If nothing else about this company is worrying you, I wouldn't be concerned.

Yeah that's not really what I'd call normal - it's legal because they are essentially operating a payroll system where you work a week in hand. Normally where this in place it goes with weekly payroll, they are operating payroll monthly while also having a "week in hand" - That's highly unusual in salaried environments - it's done for the reasons they say but is much more common in seasonal and retail environments where they have high staff turnover and are looking to avoid having to try and recover monies owed from staff.

Personally I would consider it irritating but wouldn't go so far as to call it a deal breaker. Assuming the company itself doesn't fold you'll be getting that money one way or another eventually but it's poorly worded (which would probably assist you if you ever came to a legal dispute over it in the future).

I've been in the UK job market for about 30 years (school/uni/work).

This used to be relatively common, it's called Payment in Arrears

As they say when you start you miss the first pay as you were paid a pay period behind (and if you were salaried that could be a month's pay) and got it after you left.

Now usually this wasn't really an issue as everyone did it, so at your very first job you'd have to survive pay 1, but as long as you moved directly from one job to the next you'd never be without (the first pay you didn't get in the new job would be covered by the last from the old job). You would have to remember the lag when you were waiting for overtime payments etc.

It could be quite fraught if you left on bad terms or there were problems in the company as you'd be waiting to see if you got your due.

Some point in the 90s companies started paying for the current period, likely as they now had software for payroll so didn't need the time to process deductions.

Likely this company is outsourcing payroll and has been hit by (as they say) people overusing holidays and leaving owing money that they cannot recover from unpaid salary without delaying payment.

So, not unusual but maybe a red flag about how the company works and the staff turnover.

As mentioned in the comments, this is something that does show up on some contracts but it is not the standard, to answer your specific question.

Due to the simple wording you provided, that may be inline to what is in your contract, it may be more or less bearable. If I were you, I'd specifically ask if this pay will go up with inflation or if it will be equivalent to a week's salary at the point where you leave the company (rather than your starting salary). If possible, negotiate for better wording of this term in your contract should the answer to these questions be positive. If it isn't positive, then you should re-evaluate whether the benefits of this position overcome this disadvantage. If you're in no position to reject, then you got no choice but to take it.

This is not really as complicated as everyone is making it. It's really a simple matter of being paid for the work you've done, plus the payroll processing time...and it works the same way in the U.S. where I work.

Let's say your employer processes weekly payroll and payday is Friday. The payroll person, or team, also works Monday-Friday, so on Monday they begin processing the pay for hours worked in the previous week. That process takes a non-zero amount of time and effort, perhaps a fair amount of effort depending on the complexity of the company. They push all the necessary buttons, send the request to accounting, who in turn has to write the checks or process the electronic payments, which will be distributed on payday (the following Friday). Hence, you receive your pay a week after the last day worked in your one-week pay period.

Same thing happens for longer pay periods. I'm paid every two weeks (and have been throughout most of my professional life). My pay comes the week after the closing of the two-week pay period, for precisely the same reason ... that payroll has been processing my pay and that of all my colleagues during the week following the close of my worked pay period.

Nothing complex or nefarious going on here, just an unfortunate choice of words by whoever explained it to you in the first place.