I used to live in Whitehaven so know the area pretty well. Unfortunately the big Albright & Wilson chemical plant shut down a few years ago, and Sellafield isn't the bottomless supply of jobs it once was (though still the biggest employer by far). Asides from Westlakes as others have mentioned, the West Cumberland hospital in Whitehaven sometimes recruits for lab-based posts, and Lilyhall industrial estate (nearer Workington) has a few employers. The college at Lilyhall (Lakes College) also has quite a few engineering and science related courses so teaching or lab assistant type posts come up there reasonably often.
Definitely worth checking out the National Nuclear Laboratory vacancies too (
https://www.nnl.co.uk/careers/current-vacancies/ )
Another suggestion if you fancy a complete career change is that the ambulance station for that area is at Lilyhall/Distington (so just outside Workington). The couple of ambulance stations in West Cumbria have struggled to recruit in the past I think, so you could see if North West Ambulance is doing any recruitment for Patient Transport or Technicians (I think Paramedics now do a 3 year Uni course before starting in an actual job).