if so, can I get the BMW one that was used yesterday or would it be a generic one (I guess they all do the same thing, but it would need to work with the BMW machine that it came with)?
Genuine canister is about £35 if you have to buy one yourself or you can buy a tyre inflator and sealant can for under a tenner that will do the job. However Motability should replace it for you.
https://news.motability.co.uk/everyday-tips/what-should-you-do-if-you-get-a-puncture-while-driving/
I bought an Airman Req Pilot for my own car a couple years back from Euro car parts. Cost about £25 for pump and sealant, with replacement sealant far cheaper than manufacturers branded stuff. £30 on Amazon currently!