Cumbria only came into existence in 1974 as Djm writes
Cumberland was north of what is now Cumbria.
Barrow and Ulverstone were in Lancashire and it was called Furness, all administered out of the County town which was Preston.
Westmoreland’s County town was Appleby but it became Kendal in 1889 which is where archive records will more than likely be kept.
BTW
Holker Hall is the ancestral home of the Preston, Cavendish and Lowther families. Hence if you find any of these names you may have hit the bullseye.