A world without Robbie would be a very different world. It is less likely to come to pass than a world dominated by Robbie and free of people. If nuclear war destroys humanity and most of the rest of life, a good bet for survival in the short term, and for evolutionary ancestry in the long term, is Robbie.
I have a post-Armageddon vision. We and all other large animals are gone. Robbies emerge as the ultimate post-human scavengers. They gnaw their way through New York, London and Tokyo, digesting spilled larders, ghost supermarkets and human corpses and turning them into new generations of Robbies, whose racing populations explode out of the cities and into the countryside. When all the relics of human profligacy are eaten, populations crash again, and the Robbies turn on each other, and on the cockroaches scavenging with them.
In a period of intense competition, short generations perhaps with radioactivity enhanced mutation-rates boost rapid evolution. With human ships and planes gone, islands become islands again, with local populations isolated save for occasional lucky raftings: ideal conditions for evolutionary divergence. Within 5 million years, a whole range of new species replace the ones we know. Herds of giant grazing Robbies are stalked by sabre-toothed predatory Robbies.
Given enough time, will a species of intelligent, cultivated Robbies emerge? Will Robbie historians and scientists eventually organise careful archaeological digs (gnaws?) through the strata of our long-compacted cities, and reconstruct the peculiar and temporarily tragic circumstances that gave Robbiekind its big break?