I think the problem is that meteorites need to be stored in tightly-controlled clean environments if they are to remain scientifically useful. It would be very costly to put a meteorite on public display without contaminating it. The NHM would be kicking themselves if they put a particular meteorite on display, only to discover they’d spoiled a specimen that subsequently turned out to be of scientific interest.
This is a dilemma the NHM faces with other exhibits too. I remember my disappointment when I visited a few years ago, to discover they’d removed all the fossilised dinosaurs I remember gaping at as a kid. Of course, what I hadn’t realised as a kid was that original fossils are far too precious to mount for display, and so the ones I had gaped at were all rather dubious plaster models.