Ukulele Zen • Peter Sellers
The delightful and useful Andy Ihnatko sent me the link to this. I didn't know the song and didn't know Sellers played the uke. But, of course, he did. Here he is playing and singing "I Haven't Told Her, She Hasn't Told Me",
Check out this link on the Ukulele Underground forums for more on the tune, including chords and lyrics.