Okay, I think I have this straight now. The fee is charged as 585 pounds, always. But it is paid in dollars so that amt changes based on the exchange rate.
Yeah, sort of. The fee is currently charged at $878 in the US. This is the equivalent of £585 a few weeks ago when they updated the visa site - this figure is set and will not be changed for several weeks, even if the exchange rate changes - so it is not a direct exchange rate for the day you apply.
Right now (today) £585 = $860. However, if you applied today, you would still pay $878 (= £597), so you would actually be paying more than £585 because the exchange rate has changed slightly since the US dollar fee was set.
If you pay on a US credit card, you will be charged $878 - no exchange of currency involved, you just pay the dollar amount directly. If you pay on a UK credit card, you will still be charged $878, but in order for the transaction to appear on the UK bank statement it will have to be converted into pounds (and possibly charged a foreign transaction fee).