It's a postcode lottery I'm afraid. My last hospital offered me no hope, with a "specialist" who put every block he could in my way. My new hospital has been amazing, with the DSN offering me a short one on one discussion about carbs counting, and I was on a pump in less than 2 months.
Generally speaking, you will need to go on DAPHNE or similar, and get the CCG signoff, so I think 3 months is a realistic average It is all dependent on support from your specialist and DSN, and they should guide you. I found that having CGM and being fully engaged with my control helped immensely. Even though I was new to the clinic, they could already see the huge strides forward I had taken so they were fully on board. Without the amazing help from my current diabetes team, I wouldn't have been this far forward, so for me, it is all thanks to them! Reading Think Like a Pancreas, Sugar Surfing and Pumping Insulin gave me the ability to show how serious I was, and to justify my position, and to prove that I was ready to take it as seriously as it needs to be taken.