Pounds to Euros

You get the best economy by using your pounds in Britain and withdraw euros in Greece

we assume you are in USA and the pounds came from dollars to start with.

The actual exchange rate is what it is, any variance during a journey is likely to be a reaction to time (ie the rate changes everywhere while you are in the air). What travellers need to be aware of is mark ups and/ or commission. These vary massively even within one country. Some places may have a flat rate charge within the transaction (as may some card providers) - in this case changing a relatively small amount of left over currency can work out wasteful, what appears to be a good conversion rate might only be so if you changed £500 or more.

Within UK never change currency at an airport (ATMs with a "Link" symbol are OK) or one of the little exchange offices near London rail stations where trains from airports arrive. If I need euros I used to buy them at a Post Office but then found that within large Tesco supermarkets the fee was smaller, then I found that by online ordering for collection at Tesco it was even cheaper- but the sealed envelope would contain high value notes that were difficult to use in small shops. Until a few years ago my Visa debit card made no foreign transaction charge so I could withdraw cash from a foreign ATM at the base exchange rate, now they have a flat rate charge so I do it for the full daily allowance and never small top ups. And you would need to know whether the ATM is charging a transaction fee of it's own.

The answer to your question really depends on how many pounds you have and how many euros you need, and what charges you would pay if using an ATM. Bear in mind that if you need to use an ATM for some euros any flat fees may mean it makes sense to get all your euros.

It would always be a good idea to spend your pounds in Britain and try to leave with as few as possible, if you have less than say £50 left over just save them for a future trip or trade with a friend who is coming later. Be aware that UK banknotes are changing to "plastic" , paper £5 are no longer valid, paper £10 will be withdrawn in spring next year, paper £20 are due to change in 2020. Scottish and Northern Ireland issued notes may have different dates. And the round £1 coin ceases to be legal tender in about 8 weeks time.

I live in the UK.
Just thought you might get a better exchange rate in a Greek bank.
The exchange rate in the UK is very poor at the moment, £1 = 1.06 euros
My bank doesn't charge me for withdrawals from the ATM.

You will find it difficult beat the interbank rate, or even the interbank rate +1% if that is what the ATM / your bank offers (eg with a visa debit card rather than MasterCard).

The more specialist forex dealers in London can sometimes shave it down to .5% over the interbank rate if you are buying enough.

1.06 though isn't the interbank rate; at the time of writing it is 1.09428. The best UK bureau de change figure today is 1.09.