B737 First Officer reported failure to request expeditious handling with abnormal speed indications resulted in a deviation.

Narrative

On arrival into ZZZ; RNAV XXL; while configuring for approach we noticed the speed tapes were showing abnormal indications. With landing gear down; Flaps 25 and engine anti-ice on; the Flaps 15 Speed bug was in the Yellow band and the Flaps 40 vREF was in the red band/stick shaker. We elected to go around at 1;000 ft. and requested vectors while we evaluating the situation. We resubmitted our performance data (including obtaining new weather). The landing data that came back looked correct; and was the same as the original data obtained. Unsure if we would encounter the same anomalies with the speed tapes again; and fuel at 5.6; we opted to divert to ZZZ1 for the longer runways. We chose to land Flaps 15 in order to keep the speeds above the red band. When we obtained performance data Flaps 15 for XYC at ZZZ1; the performance data generated speed was 146; while the FMC generated speed was 135. We chose the faster speed for landing. Once fully configured for Flaps 15 landing; the Flaps 15 vREF Speed was two knots above the yellow band and 17 knots above the red band. Captain became Pilot Flying for the landing and we elected to increase the speed slightly more to not land in the yellow band. We landed uneventfully and taxied to the gate.