In general, you must obtain a valid ticket for travel at the first possible oppprtunity. If you choose to pass up the opportunity, various things can happen. Some companies will only allow the purchase of undiscounted tickets on board. Others will issue a Penalty Fare which is twice the undiscounted fare or £20, whichever is greater. You may even be prosecuted for failing to produce a valid ticket when requested, which would result in a fine and possibly a criminal record.
Sometimes, it is impossible to obtain a ticket prior to boarding a train. In this circumstance, you can buy a ticket at the first opportunity without any penalty. This may be on board the train, or at the destinaton for example.
I use American Express and have never had a problem using it to buy a ticket on the railways.
There are apparently some Northern vending machines that don't accept Amex, but all train operating companies are required to accept it so you would probably be OK to buy on board if using one of those stations. Every booking office and every other TVM I know of accepts Amex.
It is less clear-cut if the only facility at your departure station is a permit to travel machine, as these only take coins. I would insert some coins in these circumstances and get a permit, just to be safe.