The mortgage won't make any difference at all to my own mortgage prospects as its a buy to let.
Au contraire - its viewed by future, potential lenders as just a risky as any other mortgage. I can see how you/your sister may think otherwise as, in an ideal world, the rental income would be covering the mortgage but this isn't how it works. YOU are liable for the mortgage payments, not the potential tenants and this income isn't guaranteed.
There are very good reasons why your sister can't get this mortgage on her own - It's insanely financially risky!
I know I'm labouring this point, but surely this point alone is giving you a "get out of jail free card" ?
Surely your sister wouldn't want to all but stop you from being able to buy your own home (at least for the next few years), so there's no need to spin more lies such as telling your family you've been sacked from your job. Also, wouldn't the timing look a bit strange and may cause them to get suspicious?
Please don't allow this situation to force you into revealing secrets you would rather keep hidden. There are other ways round this.
I hope all goes well. Be strong xx