Multiple users updating one project without using Git

It is not necessary to use git or something, use any repository that in place for your project. That will help not to lose the work, helps in the team.

When there are more than one user working on the same or related project artifacts, some time or the other it needs to use merge in the repository, be it a developer or qa team. The same applies here in your case as well.
Always the one who does the check-in first will be winner or does not need to do merging, but the one who needs to do check-in later have to merge. Some time it can be quite complex to do as this member may not aware what the other member has done, especially if worked on the same code/test. If require this member who does the check-in later have to connect with other member or take help to do the merge or merge(if other member is not available) and ask other member to review.

And to add to complex, ReadyAPI adds id to each test step which might be additional overhead to handle merging.

I am not sure if the above helps you in any way.

One thing that can be done :
- Member A, B checked-out the project
- Member A worked on say Suite1 and checked-in
- Member B worked on say Suite2 and have to check-in

In this case,
- Member B can have a copy of his modification aside, and just export the Suite2 into a separate xml file.
- Checkout fresh copy of the project, open it in the tool
- Import the suite xml that was exported earlier into the project and save the project
- Member B now is ready to check-in.
NOTE: above are for simple project, never used composite project.

So, it is better for the team member to make the check-ins at the end of the each day (than quite late) with a logical completion of what he/she is doing. And early bird gets benefit of no merging.