Well, meanwhile I've found and fixed the problem. It was in one of mobilejoomla.php files which has a meta tag 'noindex, nofollow'. Interestigly that meta tag was also affecting desktop www pages, not only m pages (initially I wrongly thought that www pages were not affected). When I corrected that meta tag to 'index, follow', everything got all right - now I see 'index, follow' tag in HTML code of all pages, both mobile and desktop.
I don't know how this mistake took place but it costed me a loss of all visitors for more than a month. Somehow only today I discovered the issue in Webmaster Tools and thus was able to find what was wrong. And as it is hard written in the php file, it must be affecting all sites using Mobile Joomla, not just my site. Don't know, maybe the paid version does not have this issue but the free version does