Thanks for your question
This is employer error, as
1) They failed to use the tax code offered to them (it could not have been officially given twice, as there is only one Part 3 to the P45 and this is the only part an employer can take information from)
2) They then failed to ask your wife to complete a P46, which would have allowed them to send this into HMRC, for a correct tax code to be issued, thereby preventing the continuing use of BR, which would not take the tax rate bands into account, and merely charge all income at 20%
There is no HMRC error, that would stand up on appeal, because although HMRC did fail to issue tax codes following the receipt of the P35 Annual Year return and the copy of your wife's P60 (P14) they do not work cases from the year end returns on the back of those errors, but instead your wifes' case would have been listed for review, on which those lists are never fully worked. So although clearly HMRC failed to make use of the information provided, HMRC error can only be considered, where the taxpayer themselves has contacted HMRC, and HMRC then fail to act on that.
Also HMRC will argue that your wife never contacted them, as one would have expected her to question the use of the BR tax code, and ask why tax free allowances had not been awarded, which only HMRC could have rectified. So HMRC are as much to blame as you wife (in their eyes!!)
You have taken the right course of action appealing on employer error, and there is sufficient evidence that the employer DID NOT
1) Act in good faith
2) Operate PAYE correctly
And I would expect some of the tax (if not all) to be collected from the employer.
If you need any further help with this, once a response has been received from HMRC, then do feel free to come back to Just Answer, and you can always ask for me Sam Tax, in your opening sentence.
If, in the meantime, I could trouble you to rate the level of service I have provided, it would be very much appreciated, as this ensures I am paid.