The family added Lyttle, who marked his 67th birthday in hospital, had dedicated most of his life to the medical industry and was a “vital pillar” of their local church.
“His sudden loss leaves a deep, painful emptiness within us, and serves as a profound reminder,” they continued.
“Always cherish every single moment you have with your loved ones, because it can all be taken away so incredibly fast.
“Even in our deepest grief, we are immensely proud that John’s final act on this earth was to give life to others.
“Three separate families had their lives changed forever because John gave their loved ones the ultimate gift: the chance to live longer.
“While we navigate this deep emptiness you left behind, we will honour you by living fully and loving one another just as you did. Rest in peace.”
After the crash, Wiltshire Police said it was keen to hear from eyewitnesses, as both the cyclist and the van were travelling in the same direction along Bradford Road.