Any advice on how to convert these leads?

I'm pretty new to having my own practice - I started in October.  Before this I worked in industry so please excuse any naivety on my part.

Clearly as a start up practice I'm keen to pick up clients ASAP.  The biggest problem is finding prospects, but having found them the second biggest problem comes when prospects are happy to move to me, but not yet.  Common issues are (especially at this time of year) that prospects who pay monthly have pretty much paid for a full year with their existing accountant so would prefer them to produce their year end, tax return, etc.  This is annoying but understandable. 

The double whammy is that the prospects won't tell their existing accountant they are moving (or appoint me) until the year end work is actually delivered.  In some cases many months later, during which time they've been paying the monthly DD towards the next year.  And so the vicious circle continues!  I understand why prospects do this (they are worried that their accounts/tax return/whatever will immediately go to the bottom of the list of priorities), but I need to find a way to persuade them the sign up with me as soon as the year end has passed and before the previous year end work is delivered.

How do people deal with this scenario and get prospects to sign on the dotted line?

Thanks in advance