Never bought anything from there online but when i joined they gave me a temporary card (in shop) until the proper one arrived , it took ages as well, probably about a month and they points were already on there
Im not sure about online - Go to your online account there you will find you card number in your personal options somewhere - if theres a number there then yes your points are allocated to your card and card will be sent soon.
There at teh very top you will see 'Advantage Card number' underneath should be your card number I do believe when you sign up to boots online your assigned a card number.