How much time did people have to take shelter during the Blitz in 1940-41?
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More of a guess than the actual answer...
One data point is the planes being used. The German bombers had a top flight speed ranging from around 300km/h to around 500km/h depending on the model. They were escorted by fighters and there could be dogfights before they reached their targets, so it's probably sane to think of that as a maximum approach speed.
Another data point is the extent of the radar coverage. If the map is anything to go by, and depending on the point in time, aircraft would get detected around 150km off the coast at best, and when it's nearly overland at worst.
Put another way, in the rosiest possible circumstances - that is, undetected until near UK soil, and flying full speed ahead with no RAF reception committee - a Luftwaffe bomber could theoretically be over London in a dozen minutes or so and over Birmingham in about an hour. In practice, radar would detect the aircraft a half hour earlier in both cases and the RAF would seek to intercept the Luftwaffe.
There also is a practical concern, which introduces another big unknown in this back of the envelope calculation. Namely, how long did it take for military staff to trigger the alarm systems in this or that location - after all, there's little point in having sirens run in more distant cities like Birmingham until it's established that they're a potential target.
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