American here. I generally just say "go straight" or "keep straight." Occasionally I get wordy and say "keep going straight." :) We sometimes do say "straight ahead" but we tend to use that when you're nearly at the destination: "go straight ahead and the store is on your right." To me, "straight ahead" implies that you only have to travel a short-ish distance. "Straight on" and "continue on" are not as common here in America, but I think they are more common in the UK.

Thanks Chris.

BTW, we'd be unlikely to say the "then" in your question. "Then" usually implies a sequence of events instead of a location in time: "He ate, and then he slept."

Thanks Linda.

None is incorrect. Anyone would understand any of these phrases meaning "No turn at this point". "Straight on" may be a little more likely in the UK than the USA.