Can I pay to upgrade after buying ticket with miles?
 

I have purchased a British Airways ticket, economy class, with my miles. Am I able to pay the difference in the economy ticket and the business ticket in cash in order to upgrade or do they demand that if you want business class you have to purchase the full business class fare? Thanks!

No, you cannot pay to upgrade a ticket obtained with miles. Unfortunately, you completed your transaction backwards.

If you have BA miles, you have the option of a) paying part cash plus part miles to buy an economy class ticket; b) paying fully with miles (any class of service); or c) buying a ticket and upgrading it with miles (depending on availability). So, if you hadn't already booked your ticket, you could have paid for an economy class ticket and used miles to upgrade to premium economy. The better value, however, is to buy premium economy and then upgrade to business class because the upgrade takes the same number of miles as the upgrade from economy to premium economy, but the difference in experience/value in the jump from premium economy to business is much greater than the difference between economy and premium economy.

If you don't mind paying a cancellation fee and you used BA miles to obtain your seat, you could look into cancelling your reward ticket and instead booking a ticket via option (c) above. Do not do this, though, if you used another airline's miles to get your BA ticket and don't do it before checking on BA's website to see if there is availability to "buy and upgrade with miles".

Many thanks! Wonderful answer!

Depending upon the airport, you can sometimes pay to upgrade at check-in. I did this as recently as last year, on a flight from San Francisco.

However, you cannot do this out of Heathrow.

What about on AA? I have a FF ticket for Buenos Aires next month, but not enough for first class.

You can't do it on AA. You'd also have to pay for coach or business & then use miles for an upgrade - if space is available.