Your code only needs to be threadsafe if you wish to enable the concurrent flag, otherwise your code will happily run as it had before.
I just thought it's important to note that they're not forcing you to push a code update unless you want to handle concurrent requests with a single instance.
Wow. I'm not a GAE user, so I'm not embarrassed that I didn't know this. However, I'm surprised that such a restriction hasn't come up in all the blog entries I've read about GAE over the years.
It's not a huge restriction since GAE will happily spin up dozens of instances without any intervention if your load requires it. Getting multiple concurrent requests into each instance will make things like local memory caching somewhat more useful though.