I'm writing on behalf of my dad who I believe is now a resident in Portugal, has been living there full time the last 2 years.
That would make him a tax resident of PT even if he doesn't have a resident's certificate or permit.
He has a UK Pension and several of his friends are being fined a lot of money for tax they should have paid on their income. (Fines include backdated non payments for 5 years i believe as well)
A friend of mine on a military pension (only taxable in the UK under the dual taxation treaty) got fined €150 for every year he hadn't submitted a PT tax return. If memory serves they can only apply fines and penalties for the previous 4 years.
it sounds as if he should pay income tax on his pension in Portugal and apply to not pay that tax i the UK as well?
That is how it works and what he should have been doing. UK state pension is taxable in PT, some UK Gov pensions (e.g. military, civil service, emergency services) are only taxed in the UK but must still be declared in PT.
There is a list of which UK Gov pensions qualify for UK taxation somewhere on the UK pensions website if your father has one of them.
Is there a Portugal tax calculator to see how much he should be paying or what is the calculation?
There probably is but the basic premise is the PT personal allowance is just over €4000 a year (UK about £12000) but other allowances can come into play, e.g. health expenditure but they are limited and not generous. PT tax bands are more numerous, the greater the income a higher tax rate applies.
What is the Uk process for not paying the income tax there - is it filing a tax return every year? How is the tax rapaid, can overpaid tax for previous years be claimed back?
You would have to ask HMRC as it may have changed after Brexit. Pre Brexit it was a simple as completing a form obtained from HMRC getting it stamped by the PT tax authority and sending it back to HMRC.
I suspect HMRC will not reimburse any tax paid to the UK without confirmation from the PT tax authority that the appropriate tax on that income has been paid to PT, but again HMRC can advise on the procedure.
Income tax is a minefield as so many personal variations come into play. My advice would be for your father to employ a PT tax accountant to get his tax affairs in order, they are a lot cheaper here than in the UK.