The structure no sooner is used to talk about something that happens immediately after something else.
- No sooner had I stepped out, than it started raining.
This sentence means that I stepped out and immediately after that it started raining. These two activities take place almost simultaneously. There is no real time difference between them.
When no sooner comes at the beginning of a sentence, we use inverted word order. That means the auxiliary verb comes before the subject.
- No sooner had I received her call, than I left for her place. (NOT No sooner I had received her call, …)
- No sooner had she finished one project, than she started the next. (= As soon as she finished one project, she started the next.)
- No sooner had I eaten the fish, than I started feeling sick. (= As soon as I ate the fish, I started feeling sick.)
- No sooner had they completed the work, than they demanded the wages. (= As soon as they completed the work, they demanded the wages.)
- No sooner had I gotten my bags unpacked than I realized that my camera was missing.
- No sooner had he graduated, than he was on his way to America.
The structure no sooner than is quite literary. It is not normally used in speaking. In a less formal style, we are more likely to say something like this:
They demanded wages soon after they completed the work.
I started feeling sick immediately after I ate the fish.