I wanted to use the SysTick timer for timing in my application, but even when writing a handler and trying to set it up, it doesn't seem like the interrupt is ever triggered. Does the chip even have such timer?
Unfortunately for you, we've chosed to not implement a SysTick timer in the nRF51822. This is an optional feature in Cortex-M0, and since we believe that all use cases are covered by the other timers, we chose to not implement a separate SysTick timer.
If you need a high-frequency timer, you should be able to use one of the normal timers for that, while if you need a low-power solution, you should look into the RTC timers.
Is it possible to use an RTOS using some of the other timers? My application doesn't handle much data transmission. Is it possible to disable the radio when not in use and have the RTOS in charge of the processor? My understanding is that the BLE radio has the highest priority. Does having an RTOS disrupt the radio operation?
It should be perfectly possible to use an RTOS on with the nRF51822 and S110, but it will have lower priority than the softdevice and the Bluetooth connection. It will hence not disrupt the radio in any way, but you'll still see interrupt latencies as described in the SoftDevice Specification.
Even though you apparently have already seen it, others who stumble upon this question may have use in reading this: https://devzone.nordicsemi.com/index.php/cmsis-rtos-with-nrf51822-and-s110
Note that nRF52 does have SysTick: devzone.nordicsemi.com/.../