The timing will be right for your new institution to subsidize the indexing, since it will be after you've started. But they may well have no funds available for such things, so don't rely on it.
Besides, do you really want to put your index in the hands of someone who knows little about your subject??? Someone who, for example, doesn't know the three or four different ways of referring to a particular event or person? Or who doesn't understand the controversial implications of listing x under category y? I can't imagine anyone except me (or scholars specializing in the exact same area) could possibly have done the indexing for my book. It would have been a royal mess. The indexing was a pain, but I got the hang of it pretty soon and worked my way through.
Ditto to all of this, and also to seniorscholar's points about indexing as proofreading. For these reasons, at my press, we always recommend people compile their own indexes.
If the indexing really is due at an impossible time, you could start now and get the bulk of the work out of the way. Explain the situation to the press and ask for their guideline sheet on indexing now.
That's rarely going to be possible, for the simple reason that there's no way to know what page numbers to enter until the book is typeset. In our experience, any attempt to "pre-index" results in a huge amount of replicated work. (It might be manageable if you wanted only an index of proper names, but I still wouldn't recommend it.)
But as to the original question, I don't see any reason why you wouldn't be eligible for whatever funds your new university might have to help with that book. Your book makes them look good, after all. (So do make sure your press knows to put your new affiliation in the author bio.)