I sent an informal query to a professor in UK and she replied me with her strong interest in my research works and considering me as a potential postdoc. She mentioned that she wants to apply for research grant this year and will inform me when she gets that. But she did not provide some details on the approximate time which she probably get the grant. In the meanwhile, I am a postdoc in another country and my contract will be finished within 4 months. As I need to make some plan about renewing my contract, is it appropriate to ask her to let me know about the approximate time of approving her application for grant? As a note, one of her postdocs has resigned but the remaining time of the current project is rather short. Can I make a request to join her team for recent project as well and wait to confirm her new grant in the meanwhile?
Don't be intimidated that she is a professor. It is certainly appropriate to continue your "conversation". Let her know of your interest in joining her team (possibly subject to funding) and give any constraints you might have. You don't need to say a lot, but thank her for her interest and express yours.
Ask her for any advice about what you can do at the current time if it seems like there is a possibility to work together.
In other words, write as if you are a colleague, not a subordinate.
Talking about money and timetables isn't a problem depending on how you frame it. You want to communicate the reason you want to know is because it's important to doing the job. Because it is.
If grants are taken care of already, you won't have to write them. If they're not. You will. If it's somewhere in the middle, is there a plan for what you should be doing to make the best use of the time regardless? These are things that are very reasonable for you to need to know.
I would approach it as: "In order to focus on the work I need to do for you and appropriately prioritize things, I need to understand your expectations and how they may change based on the funding situation."
And then ask more specific questions per your field and your expected responsibilities.
I'd make sure not to be demanding, pushy, only curious in passing, or apologetic as if you see yourself as "not important enough to know."
If you have a good reason to know, you have a good reason to ask.