i think many Labour supporters saw that after years of Tory rule undoing Socialst policys, like council houses, nationalised idustries, they had no choice as they say if you can't beat them join them. many won't like it but i think we could do with a bit of socialism, esp the way Society has gone in the last few yearsI suspect Corbyn's view is as close to socialism as Labour have had in many years. His nationalisation plans are undoubtedly socialist.
Blair realised that socialism in its purest form would not get Labour elected and to make a difference you have to be in power. His party was a socialist leaning party but not socialist in the strict sense, imo. Did he ever claim to be socialist whilst in power? I would be surprised, unless it was at a Labour conference and he was playing to the audience.