Generally you won't.

There are a few exceptions, namely some airports whose gate IT doesn't report back to BA centrally, and also on connections where if you miss a leg you may get the TPs/Avios, again depending on the local IT. But the 99% position is that you will not get TPs and Avios for unflown sectors. So you best call and get a refund of the taxes. It may not amount to much but it's money you are otherwise leaving on the table.