Final answer:
The Bevanites and Gaitskellites opposed each other primarily in the 1950s, representing different ideological wings within the British Labour Party. The two periods of bureaucratic expansion were the 1930s and the 1960s.
Step-by-step explanation:
The Bevanites and Gaitskellites opposed each other in the 1950s. This was a period of ideological division within the British Labour Party. The Bevanites, led by Aneurin Bevan, were on the party's left and advocated for socialist policies and maintaining strong ties with the working-class base. On the other hand, the Gaitskellites, whose name comes from Hugh Gaitskell, the party leader, were on the right wing of the party and sought to modify socialist principles to attract a broader range of voters.
The two periods of large-scale bureaucratic expansion mentioned in the options provided are the 1930s and the 1960s. During these times, governments in various countries increased their size and scope, mainly in response to economic challenges and the push for social welfare programs.