You should obtain your P45 from your old employer as soon as possible. Your new employer can then complete their sections and send the appropriate part off to HM Revenue & Customs (HMRC). Once HMRC have processed the form, they should issue you with a new tax code.
Unless the tax year in which there is an issue has finished or you’re unemployed again, it’s unlikely that you need to physically request a refund. Using your new tax code, your new employer should correct your tax deductions moving forward and will also refund any overpaid tax. So you should see an increase in your net pay when your new tax code comes through.
Feel free to contact your local TaxAssist Accountant if you would like to discuss this further.