I doubt it. We also experienced a mild and relatively snowless winter this year, but have been much healthier this year than the cold and snowy winter last year.
The three things that appear to cause an uptick in illness during the winter are: lowered Vitamin D levels due to no sun exposure, being in close contact with sick people due to cold temperatures forcing people to stay inside for long periods of time, and dried out noses that allow germs to enter the respiratory system readily.
Since it was warm this winter, less time has been spent inside this winter, so one would expect to see LESS illnesses not more. I don't see the warm temperatures this year promoting growth of germs or allowing them to live longer and infect more people, since germs are mainly transmitted indoors, where it is warm regardless of the outside temperatures.