I believe the recording you seek is "Experiments with Mice," by Johnny
Dankworth and his Orchestra. It hit number 6 on the Melody Maker
singles chart in July, 1956. (Source:
There was also a sequel, "Further Experiments with Mice."
"The Rise and Fall of Popular Music" by Donald Clarke describes this
recording as "'Three Blind Mice' in the styles of Glenn Miller, Stan
Kenton and others."
According to the All Music Guide, the recording appears on two
compilations on compact disc, "1956: A Time to Remember, 20 Original
Chart Hits" and the box set "Produced by George Martin."
I also found the song on another compilation, "Big Band Jazz - 36 Jazz Classics."
I will not identify sources for the CDs because I don't know in what
country you live. If you would like to buy one of these and have
trouble finding it for sale on-line, tell me where you live in a
request for clarification and I will see what I can find. The George
Martin box set should be easy to find, but it is expensive.
Sorry, I was not able to find a source for downloading the song.
The following query led me to the "Rise and Fall of Popular Music"
page linked above, which gave me the title of the recording.
"three blind mice" novelty styles
I hope this is a satisfactory answer to your question. If you need
any more information about this, please ask for a clarification.