I don't know about the UK, but in the US they measure unemployment using several different metrics. Some of those metrics are designed to capture things like discouraged workers. The US Bureau of Labor Statistics tracks the following metrics labeled U-1 to U-6, but generally the press only talks about U-3:
U-1 Persons unemployed 15 weeks or longer, as a percent of the civilian labor force
U-2 Job losers and persons who completed temporary jobs, as a percent of the civilian labor force
U-3 Total unemployed, as a percent of the civilian labor force (official unemployment rate)
U-4 Total unemployed plus discouraged workers, as a percent of the civilian labor force plus discouraged workers
U-5 Total unemployed, plus discouraged workers, plus all other persons marginally attached to the labor force, as a percent of the civilian labor force plus all persons marginally attached to the labor force
U-6 Total unemployed, plus all persons marginally attached to the labor force, plus total employed part time for economic reasons, as a percent of the civilian labor
I'm not sure that any of these capture your specific point, but the BLS does look at different aspects of the problem.