I've always attributed a limb "being asleep" to a pinched nerve, rather than lack of blood flow, but like the others, I'm not a physician. Regardless, I don't think there's a correlation between your suddenly elevated heart rate and your tingling arm.
OK. I especially think that if you do not normally awake anxious, and one day you start waking up anxious whenever you are laying on your side, you should ask your doctor about it. Not saying this is the case, but the elevated heart rate, and the fact you asked the question made me wonder. I had this happen to me, and still have questions about it...
I suggest using your judgement on stuff like this. Your description sounds benign alright, most likely just your arm not getting any blood. If the symptoms get worse though --- for example, let's say you wake up anxious and have chest pain whenever you are laying on your side, and your heart is racing, or some other more borderline abnormal symptom --- then I would ask your doctor. Arm not getting enough blood = obviously no issue if you have been sleeping on it, but it could be independent of other symptoms. But if anxious, or having chest pain, and/or maybe you have risk factors that havent been looked at in a while, this is not the result of lack of blood flow to the arm.
I have no specific training or experience, but I suspect the tingling you felt (felt that right?) is a call from your arm to your circulatory system that more oxygen is needed. The heart's normal reaction would be to increase the pumping rate as it does to other calls for more oxygen from muscles.