Tax compliance form?

My wife received a letter from barclays who we bank with in the UK.

Its headlined "for the purpose of taxation,i am a resident in the following countries and my tax identification number in each additional countrt is set out below"

Please tick the box if youre a US person and provide your us details below

Country                 TIN

She doesnt work or pay tax in the US, she works in the UK and tax is deducted by her employer.  Does she put her British and US details on the form?

Is your wife a US citizen or green card holder (a US Person)?

If so, she is resident in the USA for tax purposes. She would answer yes and give her US Social Security number on the form.

EDIT:
Just to add, if you don't want to sign it, don't do it. The bank is required to send the information to the IRS via HMRC anyway if there is no response to their inquiry.

She is american by birth,shes lived here since september 2016 on a spouse visa and has worked since last april.

Her company has begun deducting tax from her wages as she has passed the allowance threshold for the uk. Should she request the tax back and file for us taxes then?
Thanks

Her company has begun deducting tax from her wages as she has passed the allowance threshold for the uk. Should she request the tax back and file for us taxes then?

No, your wife must fulfil her UK tax obligation as she is resident and earning in the UK.

If the company is applying the UK PAYE system, and your wife has no other sources of income unknown to HMRC, then she may not have to file a self assessment return in the UK. You and your wife will know the answer to sources of income.

Since your wife is a USC, the US deems her as a resident of the US and all normal IRS tax obligations for US residents apply in addition to her UK obligation. Since her income exceeds UK thresholds, it is quite likely she will exceed all US thresholds for filing a US 1040 tax return. So, she has two tax obligations.

When your wife files a US return, and because she has been resident here since Sept. 2016, she will be able to offset either her UK income (FEIE - Foreign Earned Income Exclusion) or offset her US tax liability by using UK tax paid (FTC - Foreign Tax Credit). Neither offset is automatically granted, and one must file a US return in order to claim an offset.

There may be additional US reporting obligations such as 8938 (assets - like bank/savings accounts) and FBAR. FBAR is separate reporting to her tax return and must be done online. Both have separate thresholds.