Yes on Orchard parade there is the British embassy, the contact details will be on the notices outside ( big wrought iron thingy) make an appointment with the visa department and you can obtain the forms.
Except you want to go to the Australian Embassy. The answer is still the same though, in that it can be done. Check with the travel agent though, as Australia normally issues electronic visas to Brits at the time of ticketing for standard visits.