I got caught at BLL a few years ago like this when a re-booking wasn't ticketed. The BA staff sat me in a small room and put me directly on the phone to the Exec Club in the UK who apologised and said they'd immediately re-issue and would drop the £60 amendment fee by way of an apology. There was no way I was going to be allowed to board on that reservation otherwise. Ever since then I absolutely check for E-Ticket numbers every trip. I find the best way to see these is on the confirmatory e-mail.
On another re-issue only a month or so ago I did not receive the confirmatory email so I called. The agent said that ticketing had failed and thanked me for calling because it wouldn't have been picked up.
Also, when changing a booking by phone that needs to be queued off always ask the agent to put on the gold (if you're gold) queue as they often overlook this. Your re-issue is made within 24 hours, as opposed to much longer, even months later, as in the case of a ticket I amended for my daughter without status last year.