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If you mean universal coverage that is based on residency/citizenship (not tied to having a job) and where patient cost-sharing is usually low (often zero for hospital care) and/or capped, then you’re mostly looking at tax-funded “National Health Service” systems and some social-insurance systems with strong annual caps.

A few important caveats up front:

• “Universal” ≠ “free.” Even universal systems often have some copays, especially for outpatient visits, prescriptions, and dental. 
• “Low copays” is partly a value judgment; many countries keep them low via annual out-of-pocket caps. 

Countries that clearly fit “universal + not employment-linked + low copays / low at point of use”

Tax-funded, largely free at point of use (classic NHS-style)

These are residency-based systems; employment doesn’t determine eligibility.

• Denmark — residents are automatically covered; care is largely free at point of use (notably GP and hospital). 
• United Kingdom — NHS is residency-based and generally free at point of use for core services (some charges exist, e.g., many prescriptions in England). (See comparative context on NHS-style tax funding and cost-sharing.) 
• Spain / Italy / Portugal — generally NHS-style, tax-funded universal systems with low point-of-service charges for core care (copays vary most for drugs and some outpatient items). (Cross-country comparison framing.) 

Universal with modest fees but strong caps (Nordic “small copays + ceiling” model)

• Norway — universal coverage with user fees up to an annual ceiling, after which many fees are waived. 
• Sweden — universal; regions set fees, but there are high-cost protection/caps, especially for prescription drugs. 
• Finland — universal; has a maximum user charge (cap) for health care/outpatient treatment (details depend on category). 

Canada (very low/zero copays for “core” services, but gaps outside that core)

• Canada — under the Canada Health Act framework, medically necessary hospital and physician services are covered without out-of-pocket charges; however, prescription drugs, dental, vision coverage varies and is often not fully covered publicly.