The interest is normally only waived on full payment of the balance. If you do not pay off in full then the interest rate which for the BA Amex is advertised at (15.9% although it could be higher / lower so check your statement) will be charged on a daily basis on each transaction until the balance is paid off.

Transactions from a previous statement are not taken in to account if you paid off in full and the interest is not compounded. Interest is not charged on money you do not owe (i.e. before the date of the transaction / after part payment of the balance)

Assuming that you spent £800 on 28th April and your payment came out on 6th June (39 days) and assuming you have spent nothing else over that period and interest rate is 15.9% then you would calculate simple interest as follows:

£800 x 39 (days) x (0.159/365) = £13.59

(if you applied further transactions / the £800 is formed from multiple transactions on different dates you need to repeat the calculation for each day based on the balance)

Once repayment is received you will be charged residual interest on the remaining balance until you have cleared it in full. If you pay your balance off in full next month the interest will be calculated until that payment is received.

If you pay the remainder of the balance now you will clear the balance and they should not charge any further interest. But beware, if you have new transactions in this period, Amex may apply your most recent payment to the new purchases rather than residual debt from the previous statement. My advice would be to clear the balance in full, now if you can.

The interest they have charged based on the above sounds too high. Ask AMEX for a breakdown of how they have calculated the interest.

If they refuse to provide this make a complaint in writing formally disputing the amount of interest you have paid and requesting a full break down (they have 8 weeks to respond). After which you can refer the complaint to the financial ombudsman. I doubt that would happen as AMEX are normally quite good, unfortunately the call centre staff may not understand the interest calculation themselves so would need to escalate your request to get the breakdown if their standard response does not satisfy the customer.