You don't have to do anything at all unless you actually receive the book.  If you do receive it, you can mark it received via your Transaction Archive.   Until then, I would just wait.

ETA - she will receive her credit when you mark the book as received - just like with a regular transaction.

You can mark the book received from your transaction archive.  IF you want to.  You are not obligated to do so at all since you already have the book coming to you from someone else.  You need to mark the properly sent one received but not the first timed out one.

From the help docs:

• You are not obligated to give a second credit for a canceled copy of a book when you submitted only one request
  • You are also not obligated to send the book back (although this is a nice thing to do if the sender is a new member).
  • You are obligated to mark the book received that is on your En Route to Me tab on My Account.

If you do wish to give credit to Sender #1, you can mark that copy of the book received from the canceled transaction:

I missed that you had another copy of the book coming to you.  Hm.  It's really up to you whether you send her a credit.  I might offer to mail it back.

you don't have to send the credit. 

But if it is a book that you can re-list and get a credit for it would be the nice thing to do.  How many are in the system and how fast they get ordered will impact whether you can recoup the credit in a timely manner.

Maybe only send the credit when you get a credit for sending it out if you don't know if it will move fast.

Sound like she knows she made a mistake and will be OK with it since you have a 2nd copy coming.  Sounds like she might not be aware that the system automatically re-orders for you if the orginal transaction is cancelled.

It's all up to you whether you want to give her the credit or not.  I wouldn't feel guilty for not giving it to her. Even if it's a WL book you would still have to spend postage money to send the book out to get a credit for it.  Maybe not getting a credit for the book will be a lesson to her to make sure she marks books mailed.  But really it's up to you. 

Make sure you carefully check the return labels when either book arrives.  I have marked the wrong book received in such a case.  The new one will be on your accounts page; the old one on your transaction archive.

It is important that you give the second one the credit for sending the book - in other words make sure that is the one marked received.  They did it right and deserve their credit. (I suspect you know this and plan to do this by what you write)

You don't owe the first one a credit at all.  Look under the HELP Center and put in "I received two of the same book" and you'll get a lengthy explanation of it all.

The first sender also would have been getting emails from PBS reminding them to come mark the book mailed so they don't lose their credit.  The first sender doesn't indicate any sort of problem that caused her to not get back online to mark mailed, they just simply forgot.  I'd be not inclined to give them credit in that case.

Bertie, I'm not sure from reading your post if the other person forgot to mark their book mailed, or if it was you that didn't mark the book mailed?

ETA: Sorry, meant "or the person writing you"