I've never booked via phone, so it may just take them a bit of time to process it?
I always book online but hopefully your confirmation will come through shortly. Not sure when you are flying but from memory, you have until 48 hours before your flight to book a car service so if you dont hear anything today, you can always just ring them again
Do you get further confirmation/phone/text a day or so prior to pickup? I booked (and got confirmation) couple months ago. Just wondered whether you assume all will be ok.