While phrases like no body, some body, and some one have evolved into the compound words nobody, somebody, and someone, the similar phrase no one has never gone this route. A quick internet search reveals that noone is popular for a nonstandard word, which shows that the impulse to compound is strong among English speakers. Still, perhaps because of the potential mispronunciation generated by the doubled o‘s, no major dictionaries list noone as an accepted word. For now, no one is considered the only correct spelling.
Examples
Noone is common in internet comment sections and on websites with low editorial standards—for example:
I know noone that PREFERS fake pot to real pot. [comment on Secaucus New Jersey News]
Certainly, when it comes to Wimbledon, noone ever goes out in complete obscurity. [Bleacher Report]
But editorially fastidious publications use no one instead of noone:
No one on board was seriously hurt, but authorities were concerned about the wellbeing of the whale. [Sydney Morning Herald]
It’s a sad thing when no one loves you, most of all if you are a new vehicle being shunned by the buying public. [Globe and Mail]
But the fact that these companies lack an IT department does not mean there is no one looking after technology. [Financial Times]
President Obama meets with Senate leaders, but no one wants to back away from tough stances on taxes. [Los Angeles Times]