Q: Can the timing being off on my 2001 Toyota avalon cause the rpms to go up and down

Fluctuating RPM is not a typical symptom of a mispositioned timing belt. To rule a mispositioned timing belt in or out, though, check engine vacuum. If intake vacuum is 4-5 inches below spec., then that would be consistent with incorrect valve timing. Air pockets in the cooling system can cause fluctuating idle speed and that possibility is relevant of course as you had the cooling system open to replace the water pump. Other causes of RPM fluctuation at idle include vacuum leaks and EGR system faults, a faulty idle air control (IAC) valve or circuit, faults in the fuel pump control circuit, defective spark plug(s) or ignition system components, faulty fuel injector(s) or injector circuit faults, PCV system faults or a faulty ECM (not common). Often, the application of a real time automotive scanner can find the fault as actual engine operating parameters can be viewed during the "surging". I would recommend that you request a rough operation/rough idle diagnostic and the responding certified mechanic will perform the required diagnostic steps to get this resolved. If you have further questions or concerns, do not hesitate to re-contact YourMechanic as we are always here to help you.