Lucy__16 wrote:
the next problem is the MacBook isn’t letting me log into my iCloud (this old MacBook is new for me so setting it up), ....
So does that mean I’m stuck with no hope? I can’t update the OS as I’d need to login into iCloud to download it but can’t login to iCloud because the OS I’m running isn’t high enough?
No, it likely means the previous owner did not properly prepare the laptop for sale by using the steps in the following Apple article:
What to do before you sell, give away, trade in, or recycle your Mac - Apple Support
You will need to start at Step #6 where you must erase the whole physical SSD (Intel Macs Only).
If you want Mojave instead of Big Sur, then you will need to create a bootable macOS USB installer now while you can still boot this Mac, otherwise you will need access to another Mac or first install macOS 10.11 through Internet Recovery Mode using Command + Option + Shift + R, then upgrade to just Mojave.
If you don't specifically need Mojave and want Big Sur, then try booting into Internet Recovery Mode using Command + Option + R. Creating a bootable macOS USB installer is still a good idea so you have more options available in case Internet Recovery has any issues.
Keep in mind in order to downgrade the OS requires erasing the whole SSD which destroys all data on the SSD. If the seller did not do the steps in properly preparing the Mac for sale, then this is something you should do/have done to ensure there are no nasty surprises left by the previous owner. Even if the previous owner did all that, then I still recommend the new owner perform Step #6+ anyway.