Tape start sound

It sounds like a high frequency note going to a low frequency note, with a good amount of reverb.
Now, what makes the notes?
Maybe not quite right, but it could be done with a small drum, 8" perhaps.
You hear the bottom head first, so tune that to the high note.
Then you hear the top head, so tune that to the low note.
Normally, both heads would be tuned the same, to make it resonate.
Adding reverb is easy.
You can do this easily in Audacity using the sliding stretch effect. Choose a short area at the beginning of about 1 second. Put the pitch at the highest, and down to the normal pitch. Change the tempo and you will get a simulated effect of the tape recorder starting, which on playback gives you what McCartney did. You can play with the variables to adjust.

pitch shift down.jpg
View attachment Pitch Shift down.mp3