You seem to be missing the fact that Teledji cant turn Nanamo into a puppet cause that is practically what she already is. The real power in Ul'dah lies with the Syndicate. She is a figurehead and she has managed to claw back influence but the main power in Ul'dah resides with the Syndicate, not her.

Teledji is, powerwise, close to Nanamo's equal. If he had important business, politically he would be very hard to deny. Further, since the Sultansworn failed he would be perfectly at rights to demand that the WoL be handed over to the Brass Blades since they are the policing force of Ul'dah. The Sultansworn may be Nanamo's personal guards but they aren't in a position to overrule the Syndicate unless they have proof of Nanamo's safety being at risk.

Seriously, what do you expect the Sultansworn to do? Deny one of the rulers of the city? They don't have that authority. Maybe if they had Nanamo to back them up but without her Teledji has authority. They don't have a legal basis for saying no unless they have some indication that Teledji was involved in the poisoning himself. The WoL was literally carrying the bottle the poison came from in their pocket. Its practically being caught red handed.