When I turn on my (non-insulated) garage lights (LED), there is a 2-4 second delay. After 2-4 seconds, I hear a solid "click" from the light switch, and then the lights come up. The click sounds almost like the tension in a piece of metal being released. The delay only started recently, since it's started to get very cold (<0F), but I'm not sure if that is a coincidence. Do I have a serious problem with the switch/wiring, or is this just a symptom of an old switch and the cold?
It's probably just a bit of drag in the sprung toggle switch, which could be exacerbated by the cold.
with the power off , spray into the mechanism with some contact cleaner [if the pull-switch has a threaded or simple to remove cover, take it off first]; pull the switch a dozen times or so. You should be able to hear a slight difference between the on click & the off click, each of which can be separate to the 'pull' click - some switches operate by spring on the return rather than on the pull itself, but all will have a toggle mechanism so one action produces two results. Keep going until both clicks are immediate. Give it another 15 mins to be sure the cleaner has evaporated, then power back on.