Scenario: Lets say I have a REST API written in Python (using Flask maybe) that has a global variable stored. The API has two endpoints, one that reads the variable and returns it and the other one that writes it. Now, I have two clients that at the same time call both endpoints (one the read, one the write).
I know that in Python multiple threads will not actually run concurrently (due to the GIL), but there are some I/O operations that behave as asynchronously, would this scenario cause any conflict? And how does it behave, I'm assuming that the request that "wins the race" will hold the other request (is that right)?
In short: You should overthink your rest api design and implement some kind of fifo queue.
You have to endpoints (W for writing and R for reading). Lets say the global variable has some value V0 in the beginning. If the clients A reads from R while at the same time client B writes to W. Two things can happen.

The read request is faster. Client A will read V0.
The write request is faster. Client A will read V1.

You won't run into an inconsistent memory state due to the GIL you mentioned, but which of the cases from above happens, is completely unpredictable. One time the read request could be slightly faster and the other time the write request could be slightly faster. Much of the request handling is done in your operating system (e.g. address resolution or TCP connection management). Also the requests may traverse other machines like routers or switches in you network. All these things are completly out of your control and could delay the read request slightly more than the write request or the other way around. So it does not matter with how many threads you run your REST server, the return value is almost unpredictable.
If you really need ordered read write interaction, you can make the resource a fifo queue. So each time any client reads, it will pop the first element from the queue. Each time any client writes it will push that element to the end of the queue. If you do this, you are guaranteed to not lose any data due to overwriting and also you read the data in the same order that it is written.
How Python handles asynchronous REST requests