Why Britain occupied Iceland despite doubts about a German invasion

Could Germany realistically have invaded Iceland, and why did Britain occupy it first?

Even a small German lodgement was too dangerous to risk, so Britain preempted it despite the political cost and uncertain German feasibility.

Britain's occupation of Iceland was politically awkward because British forces entered without waiting for permission. German operations in Norway made the possibility of another enemy foothold difficult to ignore. Iceland had almost no military establishment, so British planners could not assume that Germany needed a large force to create a serious problem. The comparison with Madagascar illustrates the same preventive reasoning: doubts about the likelihood or viability of enemy occupation did not remove the incentive to secure the territory first.