I am calibrating the Hull-White model which involves finding the level of mean-reversion from the market instantaneous forward rate? Is the market instantaneous forward rate directly available from Bloomberg? Thank you.
"Instantaneous forward rate" is a theoretical construct. Two different models produce vastly different forwards; in fact, two different researchers using the same model may produce different forwards. So short answer is no, this is not available from Bloomberg.
Assuming you're trying to get mean-reversion params for the physical measure, you can just use a short-term interest rate (e.g., short-term bill rates) as a proxy. Assuming you're calibrating in the risk neutral measure, then use quoted instruments (bonds or swaps, depending on what curve you're fitting).