Hi... In Australia, a business name is one thing registered per state, and a company name Pty Ltd is another thing registered Australia wide. A domain name is another thing. Trade Marking a Logo and name is another thing.
Go to www.asic.gov.au and check if the Australian company has that name registered.
I wouldn't change the name if I were you. The only reason your name is not coming up is that your website is not UP yet and you haven't got any ranking.
Once you run a Google campaign like Ad-words then when you do a Google search you will be there. If you are in UK then your customers will see you ahead of the Australian company. Google is clever that way.