Which single was the first number one hit on both sides of the Atlantic? (number one in the UK and number one in the USA in the same week)

The unlikely song to have this honour is "Don't Let The Stars Get In Your Eyes" by Perry Como and the Ramblers.
It was number one in the five Billboard pop charts issued from Jan 10 to Feb 7 1953.

https://en.wikipedia.org/wiki/List_of_number-one_singles_of_1953_%28U.S.%29

It topped the UK charts on Feb 6th 1953:

https://en.wikipedia.org/wiki/List_of_number-one_singles_from_the_1950s_%28UK%29#1953

Note that the first UK number one "Here In My Heart" by Al Martino was also a US number one but by the time the UK charts started in November '52 its US run at #1 was over.

Nice work; someone may need to rewrite the record books. This is what I found:

Artist: Beatles
Title: Can't Buy Me Love
Notes: Million-seller. Rock music's first ever simultaneous UK/USA no. 1. From the soundtrack to 'A Hard Day's Night'.

http://www.everyhit.com/popup.php?songid=2385

I got to that URL by using their search page, and searching for March 1964 in both date fields (and leaving the rest of the input boxes as is).

http://www.everyhit.com/

Maybe Perry Como has the first simultaneous song, but the Beatles had the first simultaneous single?
(the A and B sides were identical for both releases on both sides of the Atlantic)

There is no discrepancy - "Don't Let The Stars Get In Your Eyes" is not a rock song.

Perry Como has the first simultaneous song, but the Beatles had the first simultaneous rock song.

(Same references as above.)