How would you analyse why conversions dropped on a website?

Work from measurement outwards, because the most common cause is not a real drop.

First, is it real?

• Check the tracking. A site release that broke a tag, a duplicated GTM container, or a consent banner change will all look exactly like a conversion collapse. Verify with DebugView and a real test conversion, and reconcile against the order or CRM system. This must be step one — investigating a measurement artefact wastes days.
• Check the date range and comparison period for seasonality, holidays, and day-of-week effects.

If it is real, isolate where:

• Is it traffic or conversion rate? Fewer users converting at the same rate is an acquisition problem; the same users converting less is a site or offer problem. This one split directs everything.
• Segment by channel — a single channel falling points at a campaign change, a ranking loss, or a budget exhaustion.
• Segment by device and browser. A drop confined to one browser or to mobile almost always means something broke.
• Segment by geography and landing page.
• Build a funnel exploration to find which step lost people. A step that suddenly leaks is usually a bug or a form validation change.

Then check what changed: deployments, price changes, campaign edits, or a competitor's move.

Note: Say explicitly that correlation with a release is a hypothesis to test, not proof — and that the fastest confirmation is often to walk the funnel yourself on the affected device.