JSA (and the equivalent element of Universal Credit) comes in two forms. If you become unemployed with a recent full record of National Insurance contributions you're eligible for Contribution-Based JSA, which is NOT means-tested.
Contribution-Based JSA is only payable for a maximum of 6 months though. Anyone who isn't eligible for Contribution-Based JSA (either because the 6-month period has expired or simply because they've not got a recent full record of National Insurance contributions) can apply for Income-Based JSA, which IS means-tested (with, for example, the income of the claimant's partner being taken into account). Savings, including things like redundancy money, are also taken into account.
For people living with a partner in full-time employment that typically means that they'll get full JSA for up to 6 months but, if they remain unemployed, absolutely nothing thereafter.
See here:
https://www.gov.uk/jobseekers-allowance/what-type-you-get
although this is possibly a better summary:
https://www.totaljobs.com/careers-advice/money-and-legal/how-to-claim-jobseekers-allowance