Taxation is a bit odd. Everybody must be tax resident somewhere. For example if you spent 2 weeks in France then less than 183 days in the UK and less than 183 days in Greece you couldn't claim to be tax resident nowhere. As far as I can tell, rather like nationality, there is nothing to stop you having dual tax residency?
I declare myself as tax resident in Greece and not tax resident in the UK. Greece obviously has no problem with that because I live here more than 183 days each year and it means I pay tax on my worldwide income here. HMRC has no problem with that because I only visit the UK for at most 5 weeks in a year, usually a lot less, e.g. 2 weeks for the past several years and this year zero. Unless I provide Double Taxation Forms from the Greek tax authority, proving I am declaring income here and it is being assessed for tax here, HMRC will continue to tax any income arising in the UK. Getting such DTA Forms out of the Greek tax authority is not easy and it is done for each individual income stream, so I only got DTA Forms for my two main pensions. My other pensions and interest remain taxable in the UK even though they are declared here and assessed for tax here. I chose for which income streams to get the DTA Forms so that the total income left taxable in the UK was less than my personal allowance. It is assessed for tax but the tax owed is zero. HMRC has no problem with me having two income streams marked as NT, i.e. Non Taxable, and the rest assessed for tax.
I have accounts with two UK banks. With one I use my Greek address and the other I use my son's UK address. When the first queries me about my tax residency, as they do from time to time, I tell them that I am tax resident in Greece and give them my Greek TIN, i.e. AFM. When the second asks me I don't fully complete the form but instead I send them a letter. I tell them that I spend some of my time in the UK and some in Greece, which is true. I tell them that when I am in the UK that is the address where I live, which is true. I tell them that I have income streams in the UK and income streams in Greece, which is true. I tell them that some income streams are taxable in Greece and some are taxable in the UK, in accordance with the DTA, which is true. I give them both TINs, i.e. AFM and NI Number, and tell them to provide legally required information to whichever tax authority they wish or both. They record me as having Dual Tax Residency.
As far as I can see the Withdrawal Agreement requires you to meet the legal requirements for residency in the country where you live before the end of 2020. What happens after that, with respect to documentation, is down to the country where you live and nothing to do with the UK.
I can see a possible problem in Greece for people claiming residency rights under the Withdrawal Agreement, i.e. I am living here for more than 183 days in every year, and also claiming tax residency in the UK, i.e. I am living here for less than 183 days in every year. Whether that problem will materialise depends on the communication between different Greek government departments so we cannot know until it happens. I certainly know of accountants, who keep their clients registered as tax resident in the UK, who have been asked by their clients if they should transfer their tax residency to Greece because of the Withdrawal Agreement and have told them not to. Whether that is because they are really clued up and know that it is not going to be a problem or because they don't really understand the implications of the Withdrawal Agreement I have no idea. Time will tell. My guess would be that the worst case scenario is that they need to transfer their tax residency when they try to get the new documentation but I am not saying that will happen. Greece seems to want us here so I cannot see it leading to people losing their residency rights.
Personally I tend to follow the rules although not necessarily in a timely fashion. I informed HMRC as soon as I moved to Greece because we were having correspondence about tax owed, or to be precise not owed but being claimed, so they needed my new address. I informed NHS that I had moved to Greece a couple of years later when I sold my house in the UK and no longer had a UK address. I informed DWP about 6 years after moving when I decided to pay voluntary NI contributions to protect my UK State Pension. I am not sure why people want to avoid telling these organisations? For example you can tell HMRC your Greek address but tell them that you intend to remain tax resident in the UK and they will be delighted. They are not going to object. Tax authorities tend to start digging their heels in when you tell them you are not going to be tax resident there.
You can't get into any trouble doing everything the way it is supposed to be done.