Get your taxes done using TurboTax

Your W-2 for 2019 won't change, since that's what you were actually paid in 2019.  If you didn't file in 2019, go ahead and do it now, there is no point in waiting, and the pay adjustments won't result in an amended 2019 return.

It's not clear what your employer is doing for 2020.  There are two common methods.

1) Your employer could adjust your salary for 2020 to compensate.  Suppose you were overpaid $10,000 for 2019.  Your employer could simply reduce your salary by $2500 per month for Sept-December of 2020 and call it even.  In that case, you were taxed in 2019 on what you actually received in 2019, and your W-2 for 2020 should report only what you were actually paid in 2020. Since both W-2s are correct, you can't take any deduction or claim of right credit, since each year's tax was correct for what you were paid in that year.

2) Or, your employer could pay your full 2020 salary, and collect the overpayment as a check from you or an after-tax deduction.  In that case, the 2020 W-2 would reflect your gross salary without adjustment, and you would be "out" the taxes.  That's why you make a claim of right credit, which I will explain later.  

For example, suppose you were hired in October 2019 at a salary of $10,000 per month but were accidentally paid $15,000 per month.  You are overpaid $15,000.  Your 2019 W-2 shows $45,000 and you pay tax on it.  For 2020 you are paid $120,000 and your W-2 shows $120,000.  Then after-tax, you repay $15,000 out of pocket.  Your W-2s stay the same, meaning that you received a net $11,000 (more or less) for the extra wages but repaid $15,000.  You are "out" the taxes, but you can recover that below.

It sounds like your employer is doing an odd strategy where they are having you repay the entire 2019 amount ($45,000 in my example), but then adding the $10,000 correct salary to your 2020 paychecks?  So in my example you would be repaying the entire $45,000, and then getting paid $150,000 in 2020?  That's a bit odd, and it exaggerates the effect of the repayment, but it doesn't really change the outcome.

So when you repay the gross amount that was previously taxed, you are "out" the federal, state, and social security and medicare taxes.  This is what you do next.

For federal income tax:

If the overpayment amount is more than $3000, you can either take a special itemized tax deduction for the amount of repayment, or you can take an IRC 1341 Claim of Right tax credit.  To claim the credit, you figure out what your tax would have been in 2019 without the extra wages, and the difference is the amount of your credit.  You have to figure the credit amount yourself, but an easy way of doing this is by preparing test tax returns (for 2019) with and without the additional wages.  Then, you manually enter the credit on Schedule 3 line 13 of your 2020 tax return, which you can only do using the desktop version of Turbotax installed on your own computer (not online).  That credit will come back to you as a refund of the amount of excess tax you paid on your 2019 tax return.  If you decide to take the special itemized deduction, it will be at the bottom of the Deductions and Credits page under the listing for "other uncommon deductions."  You are allowed to figure the tax benefit of the deduction and the credit and use the method that is most advantageous for you.  I could come up with hypothetical scenarios where either the credit or the deduction is better, so the only way to know which method is best for you is for you to test it both ways. 

For state income tax, you will have to research your state, there is probably a similar Claim of Right procedure.

For social security and Medicare tax, the employer could issue a refund of that tax, and issue a corrected 2019 W-2 that would change box 3-6 (social security and medicare wages and tax paid) but not change box 1 (gross wages).   If they don't, you can request a refund using IRS form 843.  You will need a letter from your employer verifying that you were required to repay wages and also verifying that the company will not be refunding the SS and Medicare tax themselves.  Form 843 is filed separately from your tax return and is not supported by Turbotax, you will need to do it yourself or see a tax professional. 

You aren't "out" the withheld tax, because you got credit for the withholding on your tax returns.  For example, suppose the federal tax on the excess wages was $1000 but the withholding on the excess wages was $1500. You got the first $500 back as your tax refund for 2019, and your claim of right credit for 2020 would be the $1000 tax liability.