I've tried using the join function but it doesn't seem to work. Is there some way for Illustrator to intuitively join up the three separate shapes smoothly whilst still keeping them on their own separate layers? If not possible on separate layers, I'd still like to know any other ways? Thanks for any help! I've attached a picture where red circles show the parts that I need joined up smoothly.
I believe the Shape Builder tool is probably the closest thing you'll get for what you're trying to accomplish as far as a time-saving method goes. Here's a demonstration:
Note: This will create one shape and will not allow you to keep each letter on a separate layer.
If you prefer to keep each letter on a separate layer, you can instead select all the shapes and use Divide from the Pathfinder pane then delete any pieces you don't want. However, doing so will create way more than three shapes; you'll have to manually re-merge them.
No need to use the join function, all you to do is to make sure your Smart Guides are active if not, you can activate them by pressing Ctrl+U or navigating to View > Smart Guides. The guides will help you precisely draw the path above the path.
Also try to preview your paths outline by pressing Ctrl+Y or navigating to View > Outline it will help you in proper aligning of the paths.
Below is my experience as I created three different paths and joined them inside the circle and also including the outline view.