The above image show the ancient crosses at St Bridget's, Beckermet - Saxon (l) and Anglo-Norse(r)
Detailed accounts of all the monastic houses in the former counties of Cumberland and Lancashire appeared in the introductory volumes of the original Victoria County Histories of the two counties, published in 1905 and 1908 respectively. These can be accessed below. No volume for Westmorland was published.
Monks: Dacre Abbey
founded before 731;destroyed c.875 by Vikings; refounded before 926; Parish Church of St Andrew built to the south of the site.
An article in Wikipedia gives a schedule of places, including possible sites in Westmoreland with some elements of their history and with extensive source references.
Premonstratensian Canons: Shap Abbey
daughter of Cockersand;(community founded at Preston Patrick before 1192(?)); transferred 1201 (1199), built (during the reign of Henry II) by Thomas Fitz Gospatrick; dissolved 1540; granted to Thomas Lord Wharton 1544/5
Below appear some further links to other papers relevant to the Religious History of the County