[link to original on tumblr]
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The Zombie Heaven liner notes list two different dates for the demo session (yielding "Summertime" and "It's Alright with Me") at Jackson's Studio in Rickmansworth. The chronology lists it as 19 April, but in the entry for the demo version of "Summertime" included on disc three, it lists the recording date as 29 April (both dates in 1964).
Interestingly, Odessey and Oracle was released on 19 April 1968, so for the sake of continuity, it'd be interesting if they really did record those demos on 19 April. That way, the Zombies as a recording entity both begin and end on 19 April - from the demo session for "Summertime" and "It's Alright with Me" to the release of Odessey and Oracle.
I'll have to do some more research though.