I’m feeling some envy right now. I’ve tried running…or rather, walks and runs interspersed, at least once a week or more often. But my heart rate shoots up even at a moderate pace of 5mi/h (7.5km/h) to around 140bpm or higher (I’m 50 years old). I just can’t run (actually jog) continuously for more than one minute before breaking into a walk. Otherwise, I workout everyday, mostly cardio and some strength for at least 45 minutes a day, every single day. My weight is in the normal BMI range. I don’t know if this is how it is to be.
FWIW I did it a few years back after coming back from an ankle fracture that had stopped me doing any exercise for 6 months. Looking at my Week 1 Day 1 activity on Strava my HR was ~105bpm when initially walking, but then up to 160bpm when running for a minute, and only dropped to ~130bpm when walking for the next minute. This see-saw repeated; up to ~160bpm when running, down to ~130bpm when walking. At the end it dropped from 160bpm to 115bpm over the course of the final 5 minute "cool down" walk. There's quite a "lag" between exercise intensity and HR.
My point is that if you're not regularly running then your CV system is inefficient, and so even walking may put your HR up quite high. If you stick at it it should improve.
I'm 47 and my HR when I go for a run is ~170bpm. If I go for a gentle run I try and keep it below 150bpm, but when I was just getting back into it I'd struggle to keep my HR under 165bpm even on a "gentle" run. As you get fitter (or lighter) you can do the same kind of runs with a lower HR than before. When I really want to push it I can average 180bpm-190bpm for a 5k (although it feels grim at the time).
If you get through a few weeks of C25K and you're still having trouble running for more than a minute or so at a time then I'd go see a doctor. If you're concerned/nervous about the idea of trying C25K with a possibility that there may be something amiss then go and see a doctor about it now.