The heart rate definitely raises slower than your pace, and while that happens your muscles work partly anaerobically. It definitely can take more than a minute, and more when you start from stand-still (which it looks like based on your pace chart).
Yes HR can take quite a while to increase, depending on conditions. Why HR shouldn't really ever be a workout target except for at the END of an interval or with some known midway situations or in steady state workouts. Like for me, I know if I am doing 10min Threshold workouts... i should be around 163-166bpm a few minutes in (with 10minute warmup before) and likely high 160's at the end of first interval... but last interval will likely hit 173 or so at the end of interval (and hit 166-168 at midpt). But day to day variations, cafeine, fatigue, time of day, heat... can all effect HR and shift it +/- 5bpm pretty easy. Also work really well to target HR on easy endurance runs where you want to keep it in a window of say 140-150... but again, would want to keep it around 138-145 early in long/easy run...but it might drift higher (at same effort/pace) later in run 144-150
