Wim pretty much summed up the most important points, but just to elaborate a bit further:
Doubling instruments are listed at the beginning of a part in their entirety, but there is no reason to do that in the order in which they appear in the piece (it will still be "Oboe / English Horn" even when the first entry is with English Horn). Also, even if you list doubling instruments in the header (see below), it would not make much sense to list them in the order of their appearance. Just imagine that you have a piece with just flute doubling piccolo, but that there are frequent instrument changes within a cue. Your header would read something like this: "Cue 12 - Flute, Piccolo, Flute, Piccolo, Flute, Piccolo, Thuringian Nose Flute, Piccolo, Flute".
The full list at the start and, if present, the full listing in the header are primarily good for one thing only: letting the players know which instruments they have to bring / unpack. This sounds trivial, but as you certainly know from your own experience as musical director, a missing instrument can be devastating to the success of a rehearsal.
But when actually playing (or even at the start of a cue) no musician will possibly ever look up to the header or to the list at the begin of the section to know which instrument is to be played next, first of all because they would have to leave the stave for that and they might lose their place.
This is why your goal of having an instrument list at the beginning of each cue strikes me as a bit unusual (and frankly, contraproductive - not to speak of wasting space*). Everything that is necessary to ensure the correct instruments playing is that instrument changes are labelled carefully. If the next cue begins with another instrument, just put in "to Bass Clarinet" after the last phrase of the preceding cue. Then state the new instrument again at its entry. Luckily, these two things are managed pretty much automatically by Sibelius when you put in an instrument change.
If at the beginning of a cue (especially those that follow some dialogue) a doubling part does not have any notes, I highly recommend to aditionally state the next instrument at the beginning of the cue. All these things are easily done with stave text styles instead of system text styles (as are the various headers that you are struggling with).
The so called "running head" (Sibelius' "header after first page") primarily serves two purposes: to match any forlorn pages to the piece and the part they belong to, and to give some basic orientation regarding multi-movement works or theatrical works with many different and often separate sections.
Elaine Gould states that only the principle instrument must be listed (meaning that just "Oboe" would be okay even when it doubles with English Horn). While this is appropriate for most concert music, I agree with Wim that there are circumstances - like a musical with the usual quadrupling (or more) of wind instruments - where it can have its advantage to also list all instruments in the header. It should be noted that this still has to be done reasonably: for example you would probably not do that with a percussion setup even if you go that route with the wind players. But here too only a simple list of all instruments needed for the whole piece (or maybe act) is necessary, there is no real use to have sub-lists for every cue. If the header becomes too long, it might be good to use (unambiguous) abbreviations.
Taking all this into account, your header maybe could look something like this:
Title of the musical - Reed 1: Cl. (Bb), Cl. (Eb), B.Cl., ASax, TSax
Act One, Scene 3, Cue 12
Everything except for the act/scene/cue indication would be easily and almost completely automatically done with wildcards.
One last thought about the concept of "current instrument" that you brought up. As far as the system barline is concerned, there actually is something like a "current instrument", because the system barline stands for one particular moment of time in the piece. But what you want is to have a list of instruments that is referring to a whole passage, which would make things more complicated (if there weren't simpler methods for such problems, see above). By definition, in a section that contains instrument changes, there can be a "current" instrument for a certain moment, but not for the section itself.
Alex
* by the way: do I understand correctly that you plan to have each cue begin on a new page in the parts? That too sounds like a huge waste of space to me, an even bigger one than putting in those subsequent instrument lists. There is no reason why you should go with anything more than a system break (together with the approriate double or final barline), except maybe that this could complicate the header labels for orientation when you have several cues on one page - but there are solutions for this, as shown a few posts above.
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