Had the P45 from your previous job included the arrears of pay, this problem would not have occurred so it really is down to your previous employer. I cannot explain what went wrong in the tax office's attempt to get any tax overpayment repaid to you via the payroll of your current employer without seeing exactly what they did so there is no point in speculating but they are only as good as the information given to them.
You need the P11D from your previous employer as well as from the current one as you had a company car with the old employer. If I were you, I'd register for self-assessment by completing a form SA1 which you can find here
and when you have been issued with a tax reference number (UTR), register to complete your tax return online here
. There are some useful notes here
By the time you are set up to complete your tax return, you may have your P11Ds. If not, complete and submit it anyway with estimates for your medical and car benefits. You can use the calculator here
to work out your car benefit. Hopefully, you will receive any tax repayment you are entitled to automatically but it could be selected for a security check which will delay it. You will need to amend the return when you have the final P11D figures.
If you don't register for self-assessment, you will have to wait for the P11D information to reach the tax office and then call them to ask them review your tax position for 2013/14 and to issue a P800 tax calculation and your tax overpayment. As the P11Ds do not have to be submitted until 6 July, it will probably be September before the situation is resolved as you have alluded to.
As for your current tax code, all you have to do to have this reviewed is to call the tax office on the number here
and they will do it while you are on the phone and make any necessary amendment.
I hope this helps but let me know if you have any further questions.