As far as I can tell inheritance taxes have to be paid within a certain timescale both in the UK and in Greece. I don't know exactly what that timescale is in Greece. In the UK
it is by the end of the 6 month after the person died. In both countries interest is charged if it is not paid on time. In the UK, in the case of a house that the person inheriting intends to live in, they can pay it in 10 annual instalments. No interest is charged on the first instalment, provided it is paid on time, but subsequent ones do incur interest and currently the rate is 3.25 % but it does vary from year to year. I think now in Greece you can also pay over 120 monthly instalments but I am not sure of the interest rate. I know it is a monthly charge and it is compound so although the rate is small it can mount up.
Brexit should not make a difference. Property ownership is dealt with under international law not EU law. Inheritance tax is dealt with under national law so it can change but Brexit shouldn't be an issue.
It is true that a UK Will can be processed through the Greek courts but most Greek lawyers advise making a Greek will because it is easier. It can be in English but following a Greek format. It is is simple to do, you simply write it yourself in your own handwriting, without mistakes, and sign and date it. It does not need to be witnessed. You don't not need to involve a lawyer or a Notary. You do need to be more specific than a UK will. For example father's name and mother's name after anybody named in the will. Village, Dimos and Nomos to specify any addresses. Name and address of the Notary that drew up the contract for the property and the contract number and date. Similarly for money in Bank accounts.
Under UK law the rules depend where you are domiciled. That is not necessarily the same thing as where you are resident. Don't ask me to define it, I am not sure anybody really knows. Pick what you think is reasonable, write a justification and see if they challenge it. When my late wife died I claimed we were both permanently resident in Greece but both domiciled in the UK and wrote a single page justification for that claim. It was not challenged. For example there is no inheritance tax limit for property passed between spouses if you are domiciled in the UK, even if you are resident in Greece. Also any unused inheritance tax allowance from when property is passed between spouses can be used subsequently when the property passes from the surviving spouse to children. On the other hand the UK considers the whole estate wherever it is in the world if you are domiciled in the UK but only property in the UK if you are domiciled outside the UK. In the UK inheritance tax is a charge against the whole estate and any tax free limit applies to the whole estate.
Under Greek law only property in Greece is considered for inheritance tax, unless things have changed recently. Also the inheritance tax is a charge against the people who inherit and the tax free allowance of each person depends on how close the relationship is. It is highest for spouses and children but I am not sure of the exact amount. Above the tax free threshold the rate does not rise as quickly as in the UK.
I hope that helps.
PS As an afterthought I notice you use the word partner. If you are not married the surviving partner may have very limited legal rights and also a very limited tax exemption. I have a friend who was a Registrar in the UK for many years and was called out to perform lots of death bed marriages because unwed partners suddenly realised the legal implications of not being married. She said that it is very disturbing and urged all partners to get married sooner rather than later whatever they thought about the institution of marriage. I would certainly talk to both a UK and a Greek lawyer to find out what the consequences are for inheritance if you are not married.