First rule is to talk to your group and point out your concerns. A Gentleman's Agreement to keep bound spirits to a mutually-acceptable level will go a LONG way towards helping this out.
Once you've done that, talk about time limits on turns and emphasize that people need to have their dice pools already in hand by the time their turn comes up. Not letting people take a long time to think about what they're doing (they should have been considering that while other people were doing stuff) helps keep combat time down.
Think about asking them if they have anyone who'd like to GM. Make that person a GM's assistant - they can have a support PC (like an AD&D healbot), but their main job is to help you run NPCs during combat, and to do rules lookups while you keep the game moving.
Finally, LIMIT CHARACTER ABILITY. The bigger the dice pools and the more stuff they CAN do, the longer it'll take. Running a combat where everyone has 20+ dice pools, 2-3 specialties, and 3 IP all the time will take a while - no way around it. Running a combat where PCs can only have a single spirit active, a 12-dice pool, 1 specialty each, and 2 IPs at most is way easier. Talk about this with your group as well - some players think that being powerful is a God-given right and the GM isn't allowed to limit what they can do. Those players are objectively wrong.