All signs point to a combined pair of symbiotic creatures taking only one turn's worth of actions per turn
I'd argue that the guest and the host that both possess the template symbiotic creature (Savage Species 131-2), when combined into the lone creature, collectively take only one turn on its turn. Although evidence is circumstantial in that nothing says the combined symbiotic creature does not take two turns on its turn, taking two turns on its turn seems, to this reader, unlikely.
Savage Species on Creating a Symbiotic Creature on Attacks, in part, says
A symbiotic creature uses the host’s base attack bonus and attack forms. However, if the guest has a different attack form (such as a stinger), the symbiotic creature can make a secondary attack with that weapon. All attacks use the host’s Strength or Dexterity modifier, as appropriate. (132)
Thus the symbiotic creature treats the guest's natural attacks as its own natural attacks except they're made as secondary attacks no matter what they were beforehand. This jibes with the example bugsucker (131) that gets guest's natural attack—the stirge's melee touch attack—as a secondary natural attack rather than the bugsucker listing the guest's attack on, for example, a different line or something.
Similarly, on Special Attacks, in part, says, "A symbiotic creature can use a special attack of either the host or the guest but cannot use both in the same round…" (132). Spells are listed in a creature's stat block in the Special Attacks section (cf. couatl (Monster Manual 37-8), lillend (MM 168)), making it impossible for the lone creature that comprises both to use in the same round spells from both host and guest.
Finally, the example bugsucker gives no indication of it receiving multiple turns on its turn. No special ability says that the bugsucker has such an advantage, so it shouldn't unless… it goes to the trouble to make itself once more into two separate creature, for which there are rules described later.
This reader suspects—and this DM would rule—that when combined, a symbiotic creature is close enough to being one creature that it takes but one turn on its turn.