The dictionary of slang you link to tells us "my giddy aunt!" comes from 1919 (from the pen of W. N. P. Barbellion) and that it's an elaboration of "my aunt!".
The same dictionary also gives some variations of my aunt!: my sainted aunt!, a mild exclamation since ca. 1920; my hat!; my stars!.
This may be unrelated, but it also notes that aunt was a procuress, concubine or prostitute (C.17 - ca. 1830) and cites Shakespeare.
[Oh,] my giddy aunt! is used as a playful euphemism to avoid blasphemy and is thought to have derived from (if not quoted from) the farcical comedy Charley's Aunt by Brandon Thomas that ran for 1,466 performances on its first production starting December 1892.
Yet more variations include my sainted aunts!, my sacred aunts! (cf. saints alive!).
Dipping into Google Books, the earliest giddy aunt is in 1837's The Monthly Review:
It may be enough to state, that a country squire's daughter, noble by
the mother's side, is sent to the metrolis, and under the protection
of a fashionable but giddy aunt, who happens to be a duchess, and a
widow, proud and poor, is appointed to entrap a coronet.
There are several more actual aunts who are giddy, but the first exclamation "my giddy aunt!" is in 1894's McClure's magazine, Volume 2 by S.S. McClure, 1894:
"The-old-devil! From the mouths of babes and suckings! Kuala! Oh, my
giddy aunt!" He went off into a peal of merriment—and the girl in
the coat embroidered with silver flowers and butterflies surveyed him
The next references are from 1908, 1910, 1913, and the aforementioned 1919.