The Nationality and Borders bill is scheduled to have its second reading in the House of Lords on Wednesday 5 January 2022. From there it will move to the committee stage approximately two weeks later, continuing its journey in that place. You can follow the bill's progress on the UK Parliament website. Debates and the committee stages are televised on Parliament TV.
The lead Home Office Minister is Baroness Susan Williams.
Here's the link to the House of Lords page dedicated to this bill:
Parliament TV link to second reading debate starts at 3:00 PM Wednesday 05 January 2022: