The United Kingdom of Great Britain was still a monarchy. It may have been a much diluted and democratic monarchy compared to the French, but it was still a monarchy. The French Revolutionaries tore down the monarchy completely and replaced it with a Republic. Its fairly obvious that the people in charge of the UK wouldnt like this on several levels.

a/ If it succeeded in France it could succeed in the UK, and this threatened the current monarchy and social structure. This was revolution, not evolution. Almost all sociopolitical structures were altered by it in France. The structures that had the power.

b/ It got brutal, just on a humanitarian level the execution of many people with a minimal level of just trials and the general mob rule was anathema to many.

This doesnt mean that many people in the UK werent in favour of it at first, anything that broke French power was good. So too, the American Revolution was clearly a strong sign that full democracy could work, perhaps this French one was sign of sweeping social changes that would assist most people. Revolution was in the air.

But in the end the brutality of the new system (that replaced a fairly equally brutal old system) altered general perception about any benefits of having a revolution. The idea of social change was good, but not if it destroyed society and brought death and mob rule to the streets.

Social revolutions are not deterred by the English Channel so the fear wasnt about a physical process. Being an island gave no guarantees of safety.

Much of the literature of the time emphasis the brutality of the new system, suggesting that this was of prime importance to many peoples concerns.

This is quite a complex subject, as many are, and is not anywhere as simplistic as 'cos they feared it would spread to the UK'. Many people wanted it to, many people opposed it. In the end it was suppressed and lacked the popularity to succeed.