See the gov.uk page for the stream conditions (red/yellow boards etc). The previous more user-friendly and informative page is no longer available.
For the flow rate see Gaugemap. This shows the volumetric flow just downstream of The Anglers; note that this is not always quite the same as the linear flow at the surface, which is what affects racing boats.
For river levels, see the gauge below. The river starts to lap over the landing-stage when the river level as shown on this page is above 3.40m and above 3.45 m you will get your feet wet when boating. You can also look at other local gauges; the river is over the top of Desborough island when the level at Shepperton Lock downstream is above about 3.35 m (there is 53 cm difference in datum between the two stations).
This information is also on the open government website at https://flood-warning-information.service.gov.uk/river-and-sea-levels?location=KT12%202JA – just click on the pins around the Club to see the river levels at Walton and at Shepperton and Sunbury Locks. This page also tells you if there are any flood alerts or warnings in place for the area.
To check the visibility (fog, mist) or other visual signs go to the webcam – see http://www.waltonrowingclub.co.uk/forums/topic/webcam-is-live/ for instructions (you must be signed in to follow this link).
Of course this information is not enough on its own to say whether the river is rowable, but it will often help.
See also the Met Office and other weather pages for weather forecasts.