00 61 from UK, and drop the leading 0 from the Australian number.
On a GSM mobile phone, dial +61 (including the plus symbol), followed by the Australian number without the leading zero.
Incidentally, you dial the Australian mobile phone this way, wherever on earth the mobile is physically located (Australia, UK, or anywhere else).
(The plus sign means "insert your international access prefix here." From a GSM mobile phone, you can enter the number in full international format, starting with the plus sign. The most common prefix is 00, but North America (USA, Canada, etc.) uses 011, Japan uses 010, Australia uses 0011, and many other countries use different prefixes.)
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In most cases, the phone number of a mobile phone remains the same. If you want to dial an Australian phone you use the same method no matter where the phone is physically located. You would dial the regular Australian phone number.
The country code for the USA, Canada, and certain Caribbean-ish islands is +1. The plus sign means "insert your international access prefix here." The default prefix from Australia is 0011. Mobile numbers in the US are indistinguishable from landline numbers; they use the same geographic area codes.
The telephone country code for Australia is +61, and you must also drop the trunk prefix 0 from the Australian domestic telephone number. For example, Sydney's area code 02 becomes +61 2.
On a GSM mobile phone, just enter the number in international format, beginning with the plus symbol. From the UK (and anywhere in Western Europe), you can also substitute the prefix 00 for the plus symbol when dialing a landline phone.
00 61 4
00 is the international access prefix from the UK
61 is the country code for Australia
4 is the code for mobiles in Australia, omitting the trunk prefix 0