How long does it take for an igneous rock to become a metamorphic roc?

There is no set answer to this.

This has been asked before. See: How long does it take for metamorphic rocks to form?

It all depends on the geological context. In case you get answers skewed to the long timescale, so I supply here examples on the shorter timescale.

A basaltic flow on the seafloor starts reacting with seawater as soon as it spills on the ocean floor. The filling of amygdules by zeolites is considered an example of the lowest-grade of metamorphism. Geothermal fields and hydrothermal vents at mid-ocean ridges are sites where active metamorphism can be observed in very young igneous rocks that are modified by reaction with seawater.

A lava flow bakes any rock it comes in contact with. The texture of some rock compositions is modified much faster than others. Deep-sea drilling expeditions have come across sites where fairly young intrusive rocks have dolomitized the overlying limestone, and zeolite minerals have grown in the pore space of sandstones.

In the examples above, the role of water is critical to reactions happening relatively fast because water provides a medium for mineral dissolution and reprecipitation.

The effect of a meteorite impact is a case of ultra-fast “shock metamorphism” on any rock that is modified but not melted. Orthopyroxenes in meteorites have been found that are shocked into an ilmenite-like mineral (akimotoite). Ultra-dense polymorphs of quartz (e.g. coesite) can also form on impact.

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