It rains a lot here, and roofs don't schedule their failures around forecasts — so this question matters practically. The honest answer is "it depends on the job", and the dependable rule is this: rain stops work that needs dry surfaces or bonding, and makes work that needs sure footing a judgement call.
What can happen in the rain
Emergency make-safe work — sheeting, temporary weatherproofing, securing loose tiles — happens in rain routinely; that's rather the point of it. Gutter clearing, inspections from ladders or towers, internal loft protection, and assessment visits are all wet-weather-compatible. Light drizzle rarely stops general repair work either, within the limits of safe footing.
What can't — and shouldn't
Anything bonded or bedded: mortar work (ridge bedding, repointing) washes out; torch-on felt won't bond to wet decks; GRP resins and membrane adhesives need dry surfaces, full stop. Stripping a roof into rain is malpractice — you never open more roof than you can close dry. And high wind trumps everything: no responsible roofer works a roof in a gale, whatever the rain is doing.
The useful tell
Ask a roofer whether they'll do your mortar or flat-roof job in tomorrow's rain. The one who says "no — and here's why" is telling you how they'll treat every invisible detail of your job. The one who says "no problem, we work in anything" is selling you a defect with a delivery date. Weather delays are not poor service; they're the service. More on the seasonal version in can roof work be done in winter?