Huduma Center do say that they will send you an SMS when your certificate is ready but they do not send it. If you want to know whether your birth certificate is ready or what stage it is at, then visit Huduma Center for a follow-up.
It is advisable to take the form to Huduma center after application if you want the process to move faster. Sometimes it takes time for them to process applications made via eCitzen portal without a follow-up.