Upvote:6
Since you and your wife are British citizens, your second son is also a British citizen (in addition to Australian, so a dual citizen). So you need to register your second son as a British citizen, and then you can obtain a passport for him.
See the following form: Application to register child under 18 as British citizen: form MN1
If you already have a passport for him, then no other action is required. The fact that he is also an Australian citizen is no problem. Bring both passports for him. See I have two passports/nationalities. How do I use them when I travel? for more information. Your son's case of dual citizenship is super simple and will not surprise anybody (airline agents, border officials, etc).
Also, even if your son were to have only Australian citizenship, Australians can visit the UK without a visa for up to six months, and the Schengen area (most of Europe) for at least 90 days. So no problem either way. See Visa requirements for Australian citizens for more information.