How do I find the calibre for a BSA Airsporter (English) air rifle?

Hi, DG,
I'm assuming you've looked at the barrel and the loading tap for caliber markings. Another avenue would be if there are any pellets with the gun, but I'm sure you'd have figured that out as well. It can only be .177 or .22, so a trip to your local Wal-mart or Big 5 (somewhere they sell pellet guns) and a tin of pellets in each caliber will answer that question. If you want to only buy one tin of pellets, get .177. Make sure the gun is not cocked! (you've mentioned you already did so in a previous post) If you can easily see the bore hole at the end of the barrel, take the pellet and hold it over the bore. if it just barely matches the diameter, the gun is .177 caliber. if it looks like it will drop in with no resistance, the gun is .22 caliber. 
Here's a link to a youtube video that shows the cocking and loading process
https://www.youtube.com/watch?v=YRkl4_ru9nI
Hope that helps!
John