To insert a header (or footer) containing information from a styled heading, use a StyleRef field, for example:
{STYLEREF "Heading 1"}
would insert the text of the first Heading 1 on the current page. If no such style is found on the current page, it works backwards through the document to find the previous one, so this shows "chapter headings" quite successfully.
I recently wanted to do something similar to you so that I could show "Heading1 - Heading2" if there was a subsection on this page, but to suppress it if there was not. Unfortunately it would give me the last Heading2 from the previous chapter if there was none on this page, which was useless. I was not able to find a way to get behind this to find which page it was returning it from (to compare if it came from before the Heading 1, for example) or any other useful tricks. It was not important enough for me to bother spending too much time on so I gave it up and just showed Heading 1.
You might be able to get what you need by having an empty heading at the top of the page with another heading directly beneath it. Mark the first one as hidden text and it might still get picked up to create a blank header on the first page of the chapter and not on subsequent ones. You may need to also look at the options for the STYLEREF field so that it picks up the first occurence on a page rather than the last (which I seem to recall is the default).