The clock starts from when you paid, I assume your application is on-line and you paid at that juncture? If you ring the Teleperormance call handlers will tell you a variety of things in relation to when the clock starts- the only consistant is thier inconsistancy. A HMPO officers told me its when they take payment.
When you apply and it gives a time frame, your case 13 weeks that is only a guideline in which they exepect to process your application – its not a contractual obligation or SLA. Yes, I would definatley try to push the application along now.
Personally,if you call I would and ask to be put through to the HMPO proper and see if a HMPO officer will expedite your application. The fact you have been waiting 13 weeks should be cause enough to be put through to the HMPO.