I've no doubt TfL and the affected boroughs are looking at this strategically, including any effect on pollution, which is certainly an ongoing crisis for the whole of London. On the High Road pollution has if anything been running lower since the bridge closure. Here's the data from the start of this year:
Looking at the most directly affected pollution monitoring sites, on Castelnau and Putney High Street, pollution has not increased, but on the whole fallen since the closure:
That's not surprising for Castelnau, but not what some people are claiming for Putney High Street.
If the reduction in pollution is considered a strategic priority for TfL, there should be no rush to reopen Hammersmith Bridge, quite the opposite.