I am going to work abroad and be paid by a US organization. They are paying into my UK bank account which charges £6 for receiving a foreign payment. I have the option of being paid in USD or another currency. Their exchange rate should be fair as it's not a commercial company. Should I ask to be paid in GBP, or USD and risk whatever exchange rate my bank has at the time of being paid?
There are probably no less than a dozen different banks that will allow you to open an account today and be happy that you are making regular deposits into that account, all without ever charging you a penny. If your bank is charging for that, they are either crazy, or ripping you off.
Now as to your question, I've found it to generally be cheaper to have transfers be in the currency of the bank receiving the money. So if your primary account is based in the UK, you're probably best off getting paid in GBP.
If you are living in US and do not have any payments in UK I think is more reliable for you to get paid in US dollars in a US bank account.
If you see how sterling pound is going down, maybe you can make more cash in the future moving your savings to UK in a cheaper currency using low charges websites like Transferwise.
I did few times from £ to € and viceversa.