Do not despair, the new cabinets have yet to be added to the BT Wholesale database used by BT Retail to determine which lines are FTTC capable. Since the exchange is already active, Basingstoke is not listed in any future rollout phase and therefore does not feature in the forward planning data available to BT Retail.
Within the next few days or weeks the cabinets will hopefully be marked active and the FTTC resellers, including BT Retail, will be able to place orders.
I recommend avoiding the BT Infinity checker, since it filters lines with low estimates, instead go straight to the BT Wholesale checker to see the full story. Check back every few days, it would be quite some time before the cabinets filter through.
Once you get the data, if you are unlucky enough to have a low estimate then try an ISP such as ADSL24, AAISP or IDnet who will be more than happy to help you get service.