You can use this API which provides a free plan for limited calls (and paid for more) to check if it works for you. In specific, you call the API with the postcode as the parameter and it returns all the related addresses.
I am not 100% sure about this, but if I remember well, they provide a free API where you can search from address to postcode and postcode to address. However, you may need to search for their documentation and how you can do this.
3) Info from Reddit
This is a copy paste from a user in Reddit who suggested something more.
You normally have to license this dataset from the Post Office for an extortionate fee. Wikileaks leaked the database in 2009, though I obviously wouldn't recommend using it for non-personal projects.
Not sure if it is allowed to post a link from Wikileaks. It would be great if a mod could clarify it.