I've not done any building projects but paying money to official organisations directly from a Greek bank account is very easy. The organisation gives you a bill for the amount you owe that contains an electronic payment code. You can use that code to make the payment, either using online banking or in the bank. I have made payments to both the tax authority and IKA using this method. I think you can also use it at an ATM but I haven't tried that.
I don't believe this would be possible directly from a UK bank account. Greek banks have payment processes set up for (virtually?) all official organisations that include the payment code while UK banks would not have that provision.
Professionals such as accountants, lawyers, engineers, architects will normally give you their bank account details, e.g. IBAN number, so you can make a transfer from your bank account to theirs, putting a reference to show what the payment is for. Technically this would also be possible from a UK bank account but you would need to think about where the currency exchange takes place and associated costs. You could also set up an account with a company like CurrencyFair
, pay money into that account from a UK bank account, convert to Euro and transfer it directly to the professional's Greek bank account.
Whether your accountant would make the payments for you is down to your particular accountant. Mine certainly does for his clients if they want him to. Obviously you would probably need to transfer the money to the accountant's bank account first.