Section 8 and 21

Tenancy Types and Management

My tenants are 4 months into a 6 month AST, and have paid no rent. I've just issued a section 21 and section 8: the former comes into force in two months (ie approx the end of the AST) and the 8 - as you know - in two weeks (followed by N5, N119, and £150 payment).
My question is: do I pursue with the N5/119 (and the associated cost) or, by the time the courts have booked a slot, will the section 21 have kicked in, and thus I would be better ignoring the S8 and following up the S21.

My aim is to get the tenants out ASAP, and claim for all rent due.

I think I understand the THEORY of all this, but am lacking the critical knowledge of how long courts take to action things.

Over to you, Great British Landlords..... (mine's a Guinness: sorry, old joke).