Question: Will I be expected to provide payslips for most job offers in the UK?
No, providing payslips is not needed almost everywhere in the world.
And any thoughts on this job given the context?
Short: run away. Longer answer below.
To negotiate a higher salary, the HR contact has requested 3 months' of most recent payslips.
They have the right to request anything. It is in your best interest to not give in to all requests. Your previous salary is confidential, between you and your previous employer.
The problem is that I don't have the paper copies and the 3rd party portal to access online payslips has been down for a month.
Maybe it is not a problem, but a benefit (in this particular case). Not having any proof might hurt you in the future (although I cannot see how).
According to the contact this is standard procedure with no other option than to accept the minimum offer.
Their internal procedures are their internal problem. You should not comply, willingly or forcefully, with everything.
I have paused the process for 2 weeks to see if I can get the pay slips from HR.
I would pause the process forever. Read below for a longer explanation.
Even getting paid on what I have requested would mean taking a slight job cut but I do want to transition into Product Management. The current offer would mean a setback of £8k per annum.
If the deal is a good deal or not for you, is for you to decide. Sometimes it is worth it to take a pay cut, in exchange for other benefits.
I am being honest about my salary but still, I thought employers couldn't request this.
You should be honest. But giving away information which you should not give is not honesty. It is falling in the trap of trickery.
Is it a bad sign that they are asking for this?
Yes, very bad sign. (keep reading)
High profile project and an opportunity to break into Product Management.
It is for you to decide what you want to pursue, and what you are willing to sacrifice or not.
Why the deal is a total lose for you and a total win for them
If you give in and give them the slips, they know that they can force you anything in the future.
If you give in and accept the minimum salary just because you do not fall into the first trap, you still fall in a trap, and they know that that they can force you into anything in the future.
So either way, you lose.
For them, if you fall in any of the traps, then you are trapped. They will abuse you in the future even more.
If you do not fall in any of the traps, you will probably not get the job, and they will find another loser to accept it - and then take their abuse.
So either way, they win.
Therefore, the only smart thing for you is to run away. Opportunities will surely appear in the future, even if in different forms than you expect them now.
My own situation several years ago
I went to an interview at a big company. They had a very strict vision and plan about what I should do there, and the chances of promotion were explicitly irrelevant. Salary could not negotiated. The start date of the job could not be negotiated (and that was a very important thing for me then). Basically, it was a bad offer, carrying the "take it or leave it" tag. Obviously, I refused.
Now I have a job which appeared in the moment when I pretty much decided to cancel my dreams. And it is a lot better than I could hope for previously - and significantly different from whatever I planned. But I am happy.
Negotiate the following. You accept the minimum salary for a 3-6 months probation period. Agree with them a clear plan with measurable results to evaluate your success at the end of the probation period. If you pass, then you will automatically go a higher salary, decided during the recruiting process.
If they refuse even this solution, not only that you should run away, but also start telling people that they are a very bad company. They do not deserve to be on the market, having this ultimatum-based behavior.