I thought the applying process would be extremely annoying and had even requested a reference from my supervisor and written a list of the material I needed to consult, but it was extremely easy. I told them I was registered for a PhD at the University of London, showed them my college card plus my passport (non-UK) for address verification, and in 5 minutes I received my BL card for a period of 3 years.
The opening hours are not fantastic, it´s from 10 to 8 most days and until 5 on Friday and Saturday I think. Learning to use the catalogue can be quite challenging but I think there are learning sessions available. I definitively recommend to order items online in advance because it usually takes up to 70 minutes to go and get them for you, if you order online the day before you can save a good amount of time.