I would add my contribution through some original US Fender catalogues I own:
If you take a look inside the 58/59 Fender catalogue you can discover the strings specially desing for "spanish guitars":
There were only three sets availaible:
Two sets called "Mastersound Flatwound" : the # 60 "gold label" (gold plated) and the #50 "silver label" and (on an other hand) one (only) set, referenced : "Pure Nickel Wrap" # 10...
All of these sets with a G wound but with no precision about the gauge.
Now if you take a look inside the 1960 catalogue you can find the same three sets. (No change)
In the 1961 catalogue, we can always find the same type of sets but now with TWO different gauges for the "Flatwound Silver label" : Always the # 50 set and a "new" one numbered: 505 "Light Gauge"...
Then, we can find the always same choice inside the 63/64 catalogue.
The "first" strings set called by Fender :"Round Wound" appears in the 1965-1966 catalogue...
It's called : "Smooth Round Wound" and the gauge is 13-52. At this time the gauge for the Master Flatwound set is 13-54, the "light gauge" Flatwound (discribe above) : 12-50
Two remarks and I would say that I 'm totally agreed with "abstamaria" :
My english is too bad to understand EXACTLY what "wrap" means ?
But certainly not the same as : "Round Wound" (?)
Because in these catalogues we can also read:
1) The "Flatwound types" are discribed by Fender as : "precision wrapped assuring a controlled diameter for absolute evenness of tone and a perfectly smooth surface for the most confortable fingering" The special metal alloys used in these strings are far superior .../...!"
2) And the "Pure Nickel Wrap Strings" are described as :" They possess extremely fine magnetic properties, and are wrapped with pure nickel to give the strings .../..."
But nowhere, in these catalogues (before the 65/66 one) you can read that these "Wrap" Pure Nickel Strings are "Round Wound"... So:
If it was not possible to buy a "Round Wound" string set from Fender in 58/59 (no more than in 60,61,62,63 and 64...) it's an evidence that the #34346 arrived, mid 59 in England from the factory, with "Flatwound strings".(May be "Pure Nickel Flatwound" but ..."Flatwound" ?!)
Besides if you take a look to this video on Youtube :
http://www.youtube.com/watch?v=gTN9NuSj43sWe can see Cliff singing "Living Doll" with his #34346 Strat in his hands.
In the "The Complete Chronicle" book (page 20) we can read that "Cliff's fifth single Living Doll enters the charts in July 1959, the 10th. So :
1) I think (and hope you think like me ?!) that the Youtube video is after this date.
May be August or September.
2) As we can see, during this clip, the #34346" is brand NEW and if you take a look to the strings, around 1'15, it's an evidence (for me...) that the strings fitted on this Strat (at this time !) are Fender "Flatwound"...
Then after that:
Remember that Saturday Dance was recorded by The Shadows (and Hank with this 34346) the August 25th 1959.
And Lonesome Fella in September the 13th of the same year.
So, if it's an "evidence" to say that Hank used (rapidly ?) Round Wound Strings (what company ?) on this maple neck Strat, he is certainly wrong when he says that he NEVER played this 34346 fitted with "Flatwound Strings" (?!)
But ?!!!!
At least....
If you take a look to this video of Duane Eddy (certainly turned in 60) :
http://www.youtube.com/watch?v=cd1ss52UIEgAs soon as the beginning of the video, we can see a set of Flatwound strings fitted on his 6120 Gretsch.
For "Mister Twang" it appears today "curious" that he used Flatwound Strings and not Round Wound on his guitar...
But was he able to buy a set like this in 60 ? Knowing that Duane Eddy is americain and lives in america at this time !
Best regards from France
Patrice
(And excuse my english mistakes... I don't practice enough !)