You will have no problems. Just a bit of a wait.
See my
What is LLU? page. O2 LLU is Partial LLU (SMPF) so the line stays with any BT Wholesale Line Rental who allow broadband with another company. (Some don't for new connections). BT Retail allows it.
There is another possibility. If the Sky line is on BT Wholesale then they also allow it. So you could have Sky line and O2 broadband if you wanted. If the line is on full LLU and just the previous owner's Sky broadband has been ceased, you can't.
To find out, just put your phone number into the O2 checker! If it offers you Standard/Premium/Pro then you could leave the line with Sky. If it offers Access then you don't want that anyway! If it says number not recognised then it is on Sky and you need to do a Return To Donor to BT (or perhaps the Post Office which is a monthly contract).
Once you get the line back to a BT WLR supplier then you can order O2. When you do that, be sure to register with either TopCashBack or Quidco, then clear cookies, and go in through the one you have chosen to get cashback. See
MKD's costing document for details.
I get the feeling (perhaps mistakenly) getting a RTD to the Post Office may nor be as simple as to BT, but may be worth it as fairly soon O2 are supposed to offering Home Phone as well. Though several of us reckon we will stay with BT as overall although the headline price is £9.50pm inc evenings & weekends, as soon as you start adding in caller display, 1571 and such things add up.
MKD is preparing a full comparison for when the product is released, but as the currently published prices may change I don't think he is releasing it just yet.
If you are interested I can find a link to those prices.