The system doesn't come well with this - I was edging into complaints territory when my 16/17yo became eligible at the end of March as household of CEV and it took two months of being pinged from pillar to post before she was properly booked in. Then we had the same drama when her part 2 fell due
The system is not properly up and running
www.bbc.co.uk/news/health-58179302
119 will tell you that you have to go via GP, they can't arrange it for those aged under the national minimum (now 16, was 18 when I was in this situation). Unless your GP still has its own hub (many have now closed) they can't do anything.
So you have to speak to the practice manager (reception staff may well be well out of their depth) and request either/both
a) GP emails local centre to make the arrangement, checking carefully that there is a paediatric vaccinator available (not all centres have them, GP can get this wrong if they are not specific)
b) GP refers to a hospital
Your GP might have better procedures than ours did, and might know exactly how to get the jab actually in to the young teen. Detailed guidance has apparently been sent out, but only in the last day or two, so readiness to deal with enquiries might vary. You might have to ring more than once, to check they really are sorting it out for you