This is the unsettling part. What they describe is accurate and seems genuine. How could they know unless they could see through my iphone camera?
Is there any way to check if my camera has been hacked or compromised in some way?
I can't help thinking that if they had control of my phone, as their email suggests, then why not empty my bank accounts using the banking app?
Have you jailbroken the phone and installed any dodgy software on it?
If not it is highly unlikely that your iPhone is compromised (there is a reason that the UK government only issue iPhones to ministers, the armed forces etc)
If you are really concerned, then take it to the Apple store and ask for a diagnostic, (use the excuse that the battery keeps going flat or that apps keep crashing) they have the software tools to check the phone is operating as per design.
You could also erase the phone and restore it to factory settings, that would remove any unauthorised software - just don't then restore your backup. (if there is unauthorised software it will be in the backup)
However - iOS is pretty much secure by default, any software that tries to be installed has to be authorised by you either with a passcode or by biometrics (face ID or touch ID)
Meanwhile, emptying your bank account is also difficult, because even if they do have access to your phone, the banks own software still needs your passcode or biometric data to access the features.