Notifying your employer of a name change is an important step — your payroll records, pension contributions, employee ID, work email, and HR file should all reflect your new name. The process is usually handled through HR or your line manager. You provide a certified copy of your deed poll, and HR will update their records and notify payroll. Your PAYE information flows through to HMRC automatically once payroll is updated, so you do not need to contact HMRC separately in most cases.
Your employment contract is a record of the agreement between you and your employer — it does not need to be re-signed simply because your name has changed. However, it is sensible to request that HR note the name change on your HR file and attach a copy of your deed poll. Some employers issue a brief letter confirming the name change for your own records, which can be useful when updating other documents.
For practical matters: your work email address may need to change, and your employer will likely want to reissue any employee ID card or access card in your new name. If you have a company pension, the pension administrator should also be notified separately — pension providers are sometimes managed by a third party and may not automatically update when HR updates their internal records.