Hi Mike,
Whilst the assessment process of ESA can take many months, payment is usually made quite promptly. It may be that the lack of an ongoing sick-note will hold things up though so I would advise that you get another one submitted promptly. The first 2 notes of the claim must be for no more than 3 months, after that they can be for as long as your GP
likes.
This is handled by DWP, it is nothing to do with your local jobcentre, it is unlikely that anyone there would be able to tell you what is happening. A call to DWP on 0800 055 6688 might tell you when you should expect some money.
Nothing on this board constitutes legal advice - always consult a professional about specific problems