Can I ask was there any complications when you made the BA booking? Did you call or did you do it online? Also any changes done to the booking once made?
Usually for any bookings made online you will get an e-ticket pretty much straight away. You should receive an e-ticket email within a few minutes. There are really very few situations where that doesn't happen. If you had to call or perhaps use a FTV then it can get sent off for ticketing which normally happens within a few days, but sometimes doesn't. Also if it takes more than a few days before they attempt to ticket, the CVV from your credit card can fall out so they can no longer take payment.
In terms of checking you have a ticket, usually as I say you should get the e-ticket email. Also on the BA app it does show a ticket number under the flight details - if it is blank that is usually a warning sign there isn't a ticket.