Can you get charged twice if you have two cards in your wallet or an Oyster and a payment card or will it reject the transaction?

It is not possible for the reader to charge two cards at the same time.  If an Oyster card and a contactless bankcard are presented to a reader on a bus together (for instance in a wallet), the readers will normally reject them both, as it can’t be sure which card was intended to be used.  TfL are advising customers to select the card that they intend to pay with, and present it on its own to the reader.

It is important to understand that a contactless card and a card linked to a contactless payment app on a smartphone are classed as different methods of payment. So if you pay by card for one journey and by app for another you're unlikely to be able to take advantage of capping. You should also make sure you use the same method of payment to touch in and out to ensure you don't get charged a maximum fare for an incomplete journey.