There are a couple of issues here, and tax law/legislation may not be the driver. In essence, the scheme is what it is. This should be determined by the type of pension fund that has been set up, and is an issue of fact. I am not sure that the pension fund manager has the discretion to change whatever fund type has been agreed, retrospectively. This could be the fault of the fund manager, or it could be the fault of the employer, or somewhere in between. Alternatively, it could be the fault of neither, but a misunderstanding by whoever operates the payroll (which is often a third party) as to how the pension should be treated for payroll purposes.
Another way of looking at it is that, if the scheme is a net pay arrangement, (in other words, the pension contribution is supposed to have been deducted from your gross salary BEFORE tax is applied - don't ask me, I didn't make up the naming convention) then the fund manager will not, I think, have the power to augment the fund by reclaiming the usual 20% tax credit from HMRC. With a net pay arrangement, the onus on calculating the tax correctly falls on the employer, or whoever is responsible for operating the payroll, so that you end up with tax relief only through your pay packet - because you will not be entitled to get tax relief anywhere else. I am no lawyer, but if this is the scenario then it looks to me like, prima facie, your employer has a serious issue to address, and the remedy will fall under employment / contract law, rather than under tax. You will have paid too much tax, had too little net pay, and your pension scheme will be underfunded. If a payroll bureau is involved, then it may not actually be your employer's fault directly, although from whom you as employees should seek remedy is a matter for lawyers.
If it is a relief-at-source arrangement, then essentially the tax treatment is correct so far, but the fund manager has a duty to reclaim 20% Basic Rate tax relief from HMRC, and you need to reclaim any marginal rate relief to which you are entitled. (You may already be aware that generally, only those employees who are paying tax at more than the Basic Rate of 20% will be at a possible disadvantage under this scenario, although strictly there are other things like Child Benefit clawback calculations that also depend on a taxpayer's "adjusted net income" - i.e., adjusted to deduct such pension contributions).
You are also correct to note that, to the extent that a claim is to be made at the individual level by the taxpayer himself/herself in relation to a relief-at-source scheme, then the clock is ticking down to the deadline for making what is now termed an "Overpayment Relief" claim for years 2016/17 and onwards, as the taxpayer has only 4 tax years in which to make a claim - so, in this case, 5 April 2021, or a little over 3 weeks.
A couple of points here:
You need to establish - quickly - exactly what scheme you have. If you have a financial (pensions) adviser, you should speak to them urgently as they will/should be able to establish the nature of the fund very quickly. If not, then you should be able to approach the fund manager looking after your pension scheme and ask them, since I should have thought that they also have a duty to you, even if they were engaged by your employer.
If its is established that it is indeed a relief-at-source scheme, then you will each need to act QUICKLY on an individual basis to reclaim the tax, etc., etc. You cannot make the employer do this, even if you think it is the employer's fault for the misunderstanding, because the legislation is designed to support only a claim made at the individual level. Furthermore, the employer will not know who is affected, since any employee may have other sources of income - e.g., rental income, and particularly where they are letting residential property that is subject to claim for mortgage interest relief - or they or their partner might be in receipt of Child Benefit, etc., etc., to which the employer will be oblivious (it occurs to me that there may be an effect on the 'old-fashioned' childcare voucher, here, as well, in some cases).
If it is actually a net pay arrangement, then you should again be speaking to your pensions adviser but also, perhaps, to a solicitor. I am not sure you will be able to "fix" things through standard tax mechanisms.
I hope this is useful. I have deliberately left out the legislation so that it is easier to read but if you want the legislative mechanisms, then let me know.
With kind regards - and wishing you good luck,