The polls I've seen put the public at 52% broadly opposed to their actions (36% broadly supportive) back in May, and I have trouble believing droning Heathrow will improve that. Sure, they have succeeded in pushing climate change higher in the news cycle, but the secular trend was in that direction anyway, "awareness" of anthropogenic climate change was at ~95% and the government already had a Net Zero strategy that was less insubstantial than the symbolic declaration of emergency, and a backdrop of emissions having fallen steadily in the UK for a very long time. Not all news is good news either: if our most-likely next PM needs an excuse to renege on his old "I'll lie in front of the bulldozer" pledge to do everything in his power to block Heathrow expansion, it's difficult to think of better cover than arguing that he's not going to let protestors set his legislative agenda...

(Also hard to imagine that legislators in more climatically-significant countries like China are likely to see climate anarchists shutting down British airports as a sign that more action needs to be taken on emissions and not a sign that certain parts of the West have lost their minds over the subject of climate change.)