When I saw WOTW at my local theater as a young lad (circa early 1960's), I can't honestly say whether or not I remember seeing the wires. But I DO remember the vivid COLORS!
Not to mention the fact that near the end of the film when the martians war machines were destroying the world with their heat, and disintegration rays, I was so scared that I accidentally swallowed a hard candy that I had in my mouth, and almost choked!
My first experience with WotW was in the 60s too Reed, on a drive-in double-bill with Psycho (if I recall correctly, Paramount officially re-released them together like that). Neither film made me choke, but that was undoubtably the most glutonous double feast I'd seen...then or since.
Despite the drive-in venue and Eastman reprint stock, I could see the wires in WotW. Later on TV, in black and white, I could see the wires as well. Even in the 70s, when I had the pleasure of viewing an almost pristine original IB Technicolor print, I could still see the wires. As others have noted, they've always been there! However, what made them much less noticeable, especially in dye transfer form, was not merely the resolution of the image, but its gamma. I seem to recall an article on the making of this movie - maybe in Cinefantastique? - which mentioned how they tested the Fx shots with the war machines and then painstakingly hand-painted the wires to match the dioramas as closely as possible. With successive home video releases though, resolution has not only increased, but so has the overall brightness...which was the most damaging flaw in the 2005 SE DVD edition, my least favourite transfer to date. In its original IB form, WotW had a richly saturated primary red-blue-green colour palette and generally sombre brightness levels, which I think contributed to its creepier feel (naturally, it played best to me in native IB Technicolor form).
So if they just stop trying to appease modern video tastes and go back to a look that is closer to the original IB Technicolor, most of these issues regarding wire visibility, will, if not totally vanish, then certainly be minimized by a more correctly timed colour/gamma balance.