I am not sure what 'losing a turn' is. Is it identical to being lost in space and time in that nothing, even monsters, affects you or fights you? Does being sucked into a gate that appears then convert me to simply being delayed?
Edit: Here is some documentation I found in the Dunwich Horror rules which implies that being drawn into a gate nullifies and previous delays and lost turns and 'resets' it in a way while retaining any possible benefits earned for the previous cost/penalty of delay/turn-loss. It likely means the accepted answer's latter section on what to do in these more complicated situations should be modified.
A link to the Dunwich Horror rules (part of interest on page 10)
Q: If an encounter, when failed, tells the investigator he must “stay here for 2 turns and gain 2 Clue tokens” but before the 2 turns have ended a gate appears at the location and draws the investigator into an Other World, does the investigator lose the 2 Clue tokens?
A: No. The investigator immediately receives the Clue tokens when he or she agrees to meet the requirements of the encounter. If a gate opens in the meantime, the investigator is sucked through the gate and becomes delayed in the Other World. He no longer has to miss two turns if sucked through on the first – the new condi- tion (being delayed in the Other World) is in effect.