Had Bronze Yohn realized that no amount of pestering could make Lysa support Robb’s war effort, and instead led the strength of House Royce into the field and joined Robb’s forces, would the events have turned out any differently than in OTL? If so, how would they have differed?

Well, as with previous questions about the whole of the Vale, it greatly depends when this happens. Let’s say Royce swings the 6,000 men of the Lords Declarant behind them – if this had happened immediately after the Green Fork, Tywin might be caught between them and Roose’s army and/or Robb’s army (if the Valemen can slow Tywin’s march), and the war could have been ended there. 

Later on, 6,000 men are less consequential.