If I lose my credit card(s) when traveling, how do you get them replaced? Do they need an address, some form of ID (this is tricky, since it could have been lost together with the card), and how much time they need for that? Do credit card companies thought about the inconvenience of losing your credit card while on the go?
The time needed to get a replacement varies from bank to bank. Some will send a card out right away via overnight courier service, others send them by mail which could take a while, some will only send them to the address on record in which case you are s.o.l.
Likewise the procedure to get a replacement varies from bank to bank. All will take reports over the phone and most start the replacement procedure over the phone. They will check your personal factoids, likely ask you to confirm some recent uses of the card (charges) perhaps your last payment amount, etc, stuff that theoretically only the cardholder knows.
Best thing to do it record your card numbers and the customer service number and keep it someplace safe, away from where the cards themselves are. You can email that data to yourself, so that you can access from any internet cafe. Some folks create a standalone email account just for such purposes.
And before you travel, you should let the bank know you are traveling and where, which would also be a good time to ask them the best procedure should you lose your card.