Britain would have been the least likely refuge for the Bourbons simply because the British political establishment largely backed the revolutionaries until around 1792. Remember, British politicians, particularly the Whigs praised the revolution at least during its early stages. They compared it to the English Revolution of 1688, believing that parliamentary government would be the result. The April 1791 constitution was praised in the Commons. Throughout the parliamentary debates of 1790 into 1791, the views seem to be largely sympathetic. What is interesting is that there seems to have been more criticism of the revolution by this time in the United States.
After the Pillnitz Declaration in August 1791, the British government was more concerned with the plight of Turkey at the hands of the Russians and Austrians than with the French Royal Family. Britain's senior diplomat in Prussia, stated that Britain should remain neutral. Lord Hawkesbury, summed up the cabinet position when he stated "if the French monarchy is not restored, France is not likely to be our enemy, unless we provoke her to it." Indeed, the British were more suspicious of Austrian and Prussian motives to induce a counter-revolution.
The anti-revolutionary camp was still very much in the minority in Britain at least during 1791. The Whigs even applauded the new constitution in September 1791 (after Varennes), and anniversary of the fall of the Bastille was celebrated openly by some British in various cities. However, opinion was slowly beginning to change and tt seems that some politicians began claiming the king was a prisoner and not acting of his free will, despite this there was still praise for the French Government. This sympathy had evaporated after the purge of the National Assembly in August 1792.
Also, if the Bourbons chose Britain as a place of exile, they would likely be without much monetary assistance. At least in one of the Austrian lands, Savoy, Naples or Spain, they could rely on the assistance of family members who would make their exile comfortable. Keep in mind the Bourbons only chose England once Napoleon became master of Europe and were no longer welcome in Courland.