Tyne and Wear only exists only as a "ceremonial county" and describes a self-defined area in which a designated Lord Lieutenant operates.
Tyne & Wear is not a local Government area nor is it an ancient country or even a recognised city or town it has not and has never been part of any official Postal Address which are regulated by the Royal Mail.
Washy got a Newcastle Post Code because when Post Codes were introduced in the mid 1960's Washy was a small pit village that didn't have its own delivery/sorting office. Post Codes reflect the location of the main sorting office that serves a settlement it does not imply the actual location of the settlement itself. Washy's NE post code is simply an administrative quirk of the Royal Mail.
The Royal Mail also regulate post codes and postal addresses . "Tyne and Wear" has never been part of any address that has been officially prescribed by the Royal Mail . Indeed the introduction of Post Codes meant that addresses no longer had to include the name of the county or region in which the address was located. (e.g it was no longer necessary to specify in an address that Sunderland was in County Durham because the SR post code only referred to that Sunderland). Likewise the name of the neighbourhood within a city or a town (e,g, Southwick, Ryhope , Roker) do not need to be included as the numbers in the first part of a post code indicate that.
I came across a bloke on YouTube the other day who had visited Seaburn specifically to make a video about it. He described Seaburn as being a small town just outside Sunderland. This was despite the fact that he began his walk into Seaburn from the Bents car park and had failed within a few yards to see the very large sign welcoming him in to the City of Sunderland.
The mind boggles - stupid and blind.