I've heard numerous reasons cited - perhaps due to the secrecy around all things nuclear - but from what I know, there are a few reasons:
Speed of deployment
Of the three designs, the Valiant was simplest, and could therefore be built and deployed the soonest. The initial Valiant prototype didn't actually meet the requirements that had been set forth for a nuclear-capable bomber aircraft, but it was better than nothing and with the mounting threat that the Cold War might turn hot, this speed was an asset, so the requirements were changed to allow the Valiant to enter service. This would fill the gap in the UK's forces, and more advanced aircraft could follow later.
Each of the V-bombers had slightly different performance, meaning that they could perform slightly different roles.
- The Victor had the greatest payload capacity (up to 35 thousand-pound bombs)
- The Vulcan was smaller and more agile and could therefore use smaller runways, and could just about break Mach 1 in a slight dive. It could also be refuelled in flight.
- The Valiant was the simplest design, and therefore the most reliable (although it later emerged that a newly-developed and poorly-understood alloy introduced some serious design flaws)
Spreading the risk
Overall, the differences between the various aircraft were pretty minor, and the main reason that three contracts were allowed to progress to delivery was simple: the UK didn't want to put all their eggs in one aeroplane. The need for a nuclear deterrent - and a means to deliver it - was so great that it could not be jeopardised by a poorly-designed bomber, a mismanaged project, unforeseen failure of the company building the aircraft, etc. By having three aircraft from three different manufacturers, the UK greatly improved their chances of having at least one aircraft that actually worked, was delivered on time, and was able to remain in service for a reasonable period of time.