If Fisher doesn't start the dreadnought leap, someone else does - either the Japanese Satsuma or the American South Carolina-class are the most likely first 'dreadnoughts'. If for whatever reason the British don't launch the first all-big-gun warship, someone else does and the British find themselves behind. Besides, you can't say the British invented the naval race when the very cornerstone of Tirpitz's naval construction (both pre- and post-dreadnought) was to build a fleet large enough to threaten the British into neutrality (the 2/3 risk theory).
Actually, Fisher was not aiming at HMS Dreadnought, a slow all-big-gun battle wagon, but at HMS Invincible, a fast all-big-gun battle cruiser, and her sisters (see: Lambert, N.A.: Sir John Fisher's naval revolution).
That was a totally different concept, which, however, most of his contemporaries didn't understand - and rather embarked on building clumsy Dreadnought-type battleships galore.
When Tirpitz started his programme, the Royal Navy counted something like 150 capital ships. Thus, the modest German naval build-up didn't really bother the Brits. - But after Fisher had devaluated the pre-dreadnoughts, the Brits suddenly realised they had a problem, because Tirpitz - reluctantly - had also switched to building dreadnoughts - and was going to construct about 60 of them within the next decade.
Now, the Brits had one of their frequent naval scares - and invented the Dreadnought Race to catch up. However, for a race you need at least two competitors. But Tirpitz couldn't run, he was tied to the naval laws, tight funding and insufficient manpower. (When he had a window of opportunity for building three or four ships per year, the fleet became querulous, because they could barely handle two new big ships per year - and not three or even four.)