The incorrect assuption is
If only one processor exists, then threads will run one after another.
How threads are being executed is up to the runtime environment.
With java there are some definitions that certain parts of your code will not be causing synchronisation with other threads and thus will not cause (potential) rescheduling of threads.
In general, the OS will be in charge of scheduling units-of-execution. In former days mostly such entities have been processes. Now there may by processes and threads (some do scheduling only at thread level). For simplicity let ssume OS is dealing with threads only.
The OS then may allow a thread to run until it reaches a point where it can't continue, e.g. waiting for an I/O operation to cpmplete. This is good for the thread as it can use CPU for max. This is bad for all the other threads that want to get some CPU cycles on their own. (In general there always will be more threads than available CPUs.So, the problem is independent of number of CPUs.) To improve interactive behaviour an OS might use time slices that allow a thread to run for a certain time. After the time slice is expired the thread is forcible removed from the CPU and the OS selects a new thread for being run (could even be the one just interrupted).
This will allow each thread to make some progress (adding some overhead for scheduling). This way, even on a single processor system, threads my (seem) to run in parallel.
So for the OS it is not at all important whether a set of thread is resulting from a single user (or even from a single call to a web application) or has been created by a number of users and web calls.