As an Australian Citizen, your wife is generally able to visit the UK for up to six months without a visa as long as she does not plan to work or study while she's there. Visiting a critically ill relative would certainly qualify. Here's the travel advice from the Australian Government (see "entry and exit"). Unless she has some criminal or immigration history that might cause her to be denied entry, she should not need a visa (or entry clearance) and can proceed to finding airfare.
It's worth asking airlines if they offer a special fare for those traveling because of family medical emergencies. Many airlines no longer offer such a fare, or it may be more expensive than the standard fare offered online, but calling and asking before you book is a good idea. Airlines also may waive change fees in such situations, especially if you can supply proof from the hospital.
I'm sorry to hear about your situation and wish your family all the best.