Are you using different template files for the blog pages with multiple and the pages with single posts?
To use these next and previous links, you need to have to different templates, or use some kind of conditional statement, but separate files would be the simplest method.
Some basic themes might use index.php as the template for both multiple and single post pages. You'll want more than that.
You should be using a template file called single.php for your single posts. This can pretty much be a copy of index.php depending on your theme. But in this template file, you'll want to use previous_post_link and next_post_link.
In your index.php file, you'll want to include the previous_posts_link and next_posts_link.
This is basically how it should be set up most of the time. Those navigation links are always going to navigate on the same level i think. So from one page of posts to the next, or from one single post to the next. But it won't ever jump from a single post to a page of posts or the other way around without the user clicking a permalink to switch where they are at in that hierarchy, which is what it sounds like you want to do.
The only way I could see that being possible is if you can somehow force the single.php file to be the homepage of your site. You might try doing research on that.