Can you check a backpack with a sleeping bag bungeed on a flight?

Absolutely. I see it frequently. It’s considered oversize baggage sometimes (depending on the dimensions) so you’ll have to pick it up at the Oversized counter in baggage claim.
Also, depending on the airline and your status with them they may charge you extra for it. Here’s an example of what Delta would charge you, for example.

To avoid extra charges for oversize or overweight baggage, your checked bag must:
Weigh 50 pounds (23 kg) or less.
Not exceed 62 inches (157 cm) when you total length+width+height.

Should be fine – just attach it securely.

Awesome, thanks. Really wasn’t expecting affirmative answers! But I don’t want to pay excess baggage fees, I already have to check an extra and pay for that :(

You can, just be prepared for the possibility that they will unroll it and not necessarily roll it back up how you had it.

Consider if you really need to take it though… where are you thinking you’ll be sleeping where you’ll need it? Youth Hostels etc provide bedding… You could also get a sleeping bag for £20 or less where you’re heading, so if it is going to end up costing you more to bring it, it may not be worth the cost.