How do I know that my VAT return has been filed successfully?

After successful submission of your VAT return, you will see the 'Filing complete' screen as per this screenshot:

When your VAT return has been processed, you can go to your VAT return page and click on the three dots and select View Past VAT reports > VAT report Summary:

You can also log into your HMRC account directly to view the status of your VAT return.