Australia and New Zealand each may refuse entry to travellers having received within the past 10 years a conviction and sentence of 12 months or more. Such a conviction would rule the visitor ineligible for the fast-track visa-waiver/electronic travel authority schemes, in which case a visa application should be sought. There is no 'automatic' refusal of a visa from such an application but it is a possibilty.
Canada, Australia and New Zealand each has its own version of 'Rehabilitation of Offenders' type legislation but there is no direct reciprocity of these laws and that of the UK. However, since 1991 Canada has generally assessed criminal convictions obtained in the UK using the Rehabilitation of Offenders Act 1974 instead of under the Canadian Immigration & Refugee Protection Act. As a result, once a criminal record becomes spent under the 1974 Act, you cease to be inadmissible to Canada and are regarded as 'deemed rehabilitated'.
http://www.bouzanethomas.com/CriminalRehabLetter.pdf
Application for immigrant visas and non-immigrant visas (through ineligibilty for visa-waiver etc.) may require a Police Certificate.
http://www.acro.police.uk/police_certificates.aspx