You cannot travel to the UK on a driving licence of any kind because it does not prove your nationality, only proves you are allowed to drive. To enter the UK, you need to satisfy UKBA of your identity and nationality.
You cannot travel to the UK on a UK birth certificate because it has no photo and because it does not prove nationality either.
You can travel to the UK on an expired British passport, but doing this in practice can be tricky, especially if flying, it would require an airline to phone UKBA in advance and it would depend on their response whether CLA would be levied if the person turned out not to be the holder of the document. The expired passport is more than likely to be removed at the port of entry and forwarded to the Passport Office, the absolute discretion to waive this rule is down to the chief or inspector on duty. The easiest place you could travel on an expired passport is through juxtaposed ports such as Calais or Paris, where the identity and nationality can be checked at that point, where there is no CLA applied on the carrier. In the past, before 9/11, the use of expired passports and using various forms of docs such as UK driving licences/birth certificates was far more lax, there was even a period of waiving through passengers who waived an EU passport, no opening up - won't even go into the UK security consequences as a result, the person responsible for this needs to hand his head in shame for allowing it. The rules for UKBA are now clear, only expired British passports, British emergency passports, British regular passports are to be accepted, any exceptions would possibly go up to HQ and anyone breaking these rules risks serious disciplinary action. Having said all that, there is an area called COMMON TRAVEL AREA which has been in existence since Ireland's independence. If travelling from Ireland to the UK or from Jersey etc, you don't need a British passport, you can travel on docs of lesser status if Irish or British.
The only Australians who need entry clearances for the UK are ones coming here to live, as 6+ months students, on work permits, UK ancestry etc.
First issue British passport in 3 weeks on holiday? Maybe a bit of a high risk, for first born already issued with a British passport, I'd take the risk of getting back a renewal, might even get it back in just a week. Good luck.
DS1 is British other than by descent and can pass on British citizenship to all descendants, DS2 is British by descent and under current UK nationality laws, if he comes to live in the UK for a continuous 3 year period during his life BEFORE he has any children, then his children can also register as British citizens by descent if they do it before they reach 18 years old, under section 3(2) of the British Nationality Act 1981 - send him on a 3 year working holiday to the UK!!!
You will indeed need a copy of your own full UK birth certificate now to apply for DS2s British passport, you might not have needed it for DS1 but now the rules have been tightened. You can apply for it online, just make sure you do it on the right link, there are companies who charge a premium price for this and all they do is then go to the official website and do the application you could do yourself.