The Beatles first British number 1 single?

Was Please Please Me The Beatles first UK Number One?

At the end of the lengthy recording session for the single version of Please Please Me George Martin turned to The Beatles and said, 'Gentlemen, you have your first Number One!' George was rarely wrong about anything but in this case the jury is still out. 

Why is there confusion?

In the early 1960s there were three major records charts in the UK. The New Musical Express and the Melody Maker both had Please Please Me at Number One in February 1963.  But on the 'official' Record Retailer (later Music Week) chart used by the industry it only reached Number 2. 

Wikipedia expresses the current consensus as follows:

"From Me to You" was the first Beatles song to reach number one in the UK and is widely considered to be their first chart-topping song, for although "Please Please Me" reached the summit on almost every chart, it was only number two on Record Retailer's chart, generally considered to be the most authoritative for the time.

 Written by John and Paul on Helen Shapiro's tour bus travelling through Shropshire, From Me to You stayed in the top spot for seven weeks.

How did George Martin improve Please Please Me?