Tachycardia or high pulse rate can also be due hyperthyroidism, fever, low blood sugar, anxiety and panic disorder, heart
causes like sinus tachycardia and palpitations
, alcohol, coffee, effect of certain drugs etc. Hence, you also need to rule out heart causes by consulting a cardiologist and getting an ECG and ECHO.
Apart from this it is very important to keep the asthma and COPD under control with proper medications. In your case, probably intermittent oxygen therapy can also be tried.