Looking on the internet deeply has found these results:

but no sooner is the most popular phrase on the web.

but not sooner 1283,000 results on the web	but no sooner 12,250,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
Perhaps, but sooner or later there'll be other. It's not perfect yet, but sooner or later... But sooner or later you get cornered. But sooner or later you have to provide for your family. Maybe not, but the sooner you accept it, the easier your decisions will be. Maybe not, but the sooner you accept it, the easier your decisions will be. But not sooner than six months from now. But sooner or later, every man shows himself.	or request immediate issuing of the licence, in which case the competent authority shall issue it without delay but no sooner than the Monday following the lodging of the licence application. When the time comes, she and her brother must be brought together in Odessa, but no sooner. Cosmetics (including sunscreens) can be applied over the treated areas, but no sooner than five minutes after application. But no sooner has Mr Schröder left than they are at it again. But no sooner had he turned up, the Doctor was gone again, leaving Amy wondering where her raggedy man had disappeared to. But no sooner do I become indispensable than they bring in a machine to do the job!

but not sooner

1283,000 results on the web

but no sooner

12,250,000 results on the web

More popular!

Perhaps, but sooner or later there'll be other.
It's not perfect yet, but sooner or later...
But sooner or later you get cornered.
But sooner or later you have to provide for your family.
Maybe not, but the sooner you accept it, the easier your decisions will be.
Maybe not, but the sooner you accept it, the easier your decisions will be.
But not sooner than six months from now.
But sooner or later, every man shows himself.

or request immediate issuing of the licence, in which case the competent authority shall issue it without delay but no sooner than the Monday following the lodging of the licence application.
When the time comes, she and her brother must be brought together in Odessa, but no sooner.
Cosmetics (including sunscreens) can be applied over the treated areas, but no sooner than five minutes after application.
But no sooner has Mr Schröder left than they are at it again.
But no sooner had he turned up, the Doctor was gone again, leaving Amy wondering where her raggedy man had disappeared to.
But no sooner do I become indispensable than they bring in a machine to do the job!