Hello Pringles
Welcome to Poms in Oz.
You need to be talking either with the General Register Office or with the Registrar of Births, Marriages and Deaths in the district where you were born. Do not bother with the Agencies who say they can get birth certs etc for you. Half of them are just parasites and the other half are into using the information for identity frauds.
The GRO website is here:
General Register Office (GRO) - Official information on births, marriages and deaths
Records of births do not "go missing." If somebody's birth has been registered then this simply cannot happen. Therefore I am very suspicious of the story about an alleged glitch in the system, so I wonder whether you have been talking with an agent rather than with the Registrar General's office?
If there really is a problem at the GRO then it is up to them to send you an e-mail explaining what has gone wrong at their end and what they are doing about it, so explain that you need that e-mail in connection with an application for an Australian visa and insist that the e-mail is sent to you straightaway. Thump the table and insist on speaking with the Manager of the Births section if necessary.
Then do a Statutory Declaration explaining what has happened, and attach a copy of the GRO e-mail as an Exhibit to the Stat Dec. That will solve that problem and a solicitor will then charge £5 to administer the Stat Dec. DIAC will accept the Stat Dec and the GRO e-mail, but they will insist on the Stat Dec so you can save time by cracking on with that.
Best wishes
Gill