I think Shropshire Archives (at Shrewsbury) will be the place to look?
On their website it says:
We're funded by Shropshire Council and the borough of Telford and Wrekin.
In 1995 the County Record Office and Local Studies Library were brought together in purpose-built accommodation in the town centre.
From 1995 to 2003, the building was known as the Shropshire Records and Research Centre. After consultation, in 2003 the name was changed to Shropshire Archives - gateway to the history of Shropshire and Telford.
Shropshire Archives had a major refit of the public spaces in 2009. The building work means that visitors to the service now benefit from improved and more spacious public areas.
Telford & Wrekin Council say:
The library service holds microfiche copies of registers at the Community History Centre at Wellington Library covering christenings, marriages and burials.
The Index to parish registers is held at the Community History Centre, Wellington Library.
The Shropshire Collection is now available on Find My Past covering baptisms, banns, marriages and burials up to 1900. This is available with free library membership on all library computers.
