"To the W of the High Street [Bridge Road/Addlestone Road] there is first a pretty three-arched iron bridge of 1865, with stone piers and openwork spandrels in Westminster Bridge style".

So, technical terms first: 3-arched (you've got this), stone piers (the base of the 2 central columns of the bridge, built in the river - the 2 columns at either end are abutments) openwork spandrels (the wedge shaped—mostly open—space between road-level, arch & masonry)

The next point is that it's Westminster Bridge style—I can see that. Much smaller scale, 4 fewer spans, no lanterns, less decoration in general, and balustrades as parapet, rather than the trefoil-design of W'minster. W'minster built 1862, Wey built 1865.

However, this raises a Q. The guide described stone piers, and W'minster shows what they look like. Wey Bridge has some stone, but it's mostly a slightly ugly black brick. But the @SurreyCouncil sign suggests that the brick is original:

@HistoricEngland back this up "Impressed in many of the bricks forming the parapet wall is Joseph Hamblet, Oldbury, Birmingham, 1865. Grey brick piers with stone coping and dressings, cast iron arches and balustrading." https://historicengland.org.uk/listing/the-list/list-entry/1377447—think the guide may be wrong.

Both @HistoricEngland and the guide note the bypassing of the bridge. With a concrete single-span road bridge built in 1945. The "pretty" bridge is much prettier for this. Most traffic by foot, sitting at the confluence of the the Wey, near the Navigation and Thames. I like it.

The final point in this thread is part apology, part annoyance. In walking Weybridge, I've noted how many gated private roads there are. They obviously have their fans, but I don't like them, it makes it very hard to take photos. The bridge is here: https://goo.gl/maps/Vq5L9ZKBBexwxfzH9