I note that you mention a contract. Does this specify a start date or does it include a clause that says that you will only get paid once the references are verified.
You are clearly being suffering prejudice her because it would seem that they can take as much time as they want in sorting out the references and other issues whilst you must simply wait for them to come back to you at their discretion, and you can take another job in the mean time.
It would appear that you are obliged to simply hang on for them and they do not have an obligation to your needs (like income etc).
It is unacceptable that if they had problems they should canvass this with you so that you could follow up on references etc- like you have done when they eventually told you about the delay upon your enquiry.
Check the contract and see if you can say that you want the salary for the 3 months.
If this is not the case push them to start in any event but diplomatically mention ( in writing) that the delays is prejudicing your financial situation which you can afford and therefore must start immediately.
By doing this you would be putting them on sufficient notice to proceed against them.
A legitimate expectation of ongoing employment has been made and it is unreasonable for them not to have addressed the actual commencement of the employment yet.
You may argue that you have been employed and they have allocated tasks to you as yet - hence wanting all emoluments relating to the role from a reasonable time after the contract was issued to you.
GOOD LUCK