The answer is always going to be the same for any EU carrier and it has nothing to do with BA.

It will depend on the reason for the cancellation of the MAN-LHR flight. If that cancellation was an "extraordinary circumstance" then the compensation is EUR 0. If it was not, then there will be compensation.

The sole question as to the latter is how much compensation. Compensation is determined by the length of delay at the final ticketed destination. That would be 6 hours at BOS. As BOS is a Type III flight, the compensation is EUR 600.

As to the weather, aviation is a good deal more complex than what your friend may believe and have told you about what he saw looking out of a window at MAN. If he is indeed expert in meteorology and other associated air traffic issues, my apologies. If not, let us presume that his carrier did not lie.

No need for a separate thread on BA, AZ or anywhere else.