How to write an effective but very gentle reminder email to supervisor to check the Manuscript?
I will just write how I approach the matter with very busy supervisors until now. It has worked without issues and our relationships has not been impacted at all.

The best thing is to take responsibility of the submission process (if you haven't done that already). When you have either the manuscript ready to submit or after you have done the corrections on the reviewers' comments, you can send an e-mail to your supervisor telling him/her that you have done all the corrections and answered to the reviewers (if applicable) and you are ready to submit. 

Put a deadline for the submission (not the journal deadline, but yours) and ask him to send his comments, if any, by then. You are planning to submit it on that day, unless you have his comments that require big changes. 

Also ask him if he needs to change the deadline, then he can propose a new one. But don't leave it without a deadline, as it could go forever.