I am an English speaker and I have never heard this expression or use of the word 'bushed' in my entire life. If a friend said this to me I would be very puzzled. It seems like a strange choice to include an idiom that most people have never heard let alone use.
upon further googling this is far from a commonly used phrase. I for one have never seen or heard it used. IMO it would be better for DL to change the default English sentence to 'I am flat out' which I believe most English speakers would recognise.