Do I need vaccinations to travel from the UK to Singapore and Australia?

Believe so for Singapore think it Tetnus B but just ask your GP he tell you which ones you might need

No -You don’t require vaccinations unless you have come from, or have visited a yellow fever infected country within six days of your arrival I've travelled many times between Singapore, Uk, and Sydney - it's only really africa/south america you need jabs for i believe.

Its the holiday visa you gota make sure you have! Please see attached health guide from the au.gov

Vaccinations are not required for visiting Singapore unless you have
traveled to or transited an area which is yellow fever infected. Other then that, Singapore is relatively high in hygiene and health care standards. In Singapore, you are perfectly safe from water-borne or food-borne diseases. It is recommended though to take flu vaccine just to prevent from contracting any flu symptom that might ruin your stay in Singapore.

No, as UK is infectious disease safe nation, no need for vaccinations from UK to both Singapore and Australia.

IS ALL DEPEND ! ( Singapore and Australia about same requirement ).
Yellow Fever Vaccination Certificate
For entry into Singapore, a valid yellow fever vaccination certificate is required from travellers (over one year of age) who, within the preceding six days have been in or have passed through any country endemic for yellow fever.
The International Certificate of Vaccination for yellow fever is considered valid 10 days after vaccination. The certificate is for a period of 10 years.
READ THIS.
http://www.ica.gov.sg/page.aspx?pageid=95&secid=94#yellowfever
ie . Within the Preceding SIX DAY you visit Uganda .. even you flight from UK. you NEED THAT !
Those Answer said NO. may make you in very Inconvenience . if you visit those countries endemic YF for the pass 3 weeks , Immigration will DENIABLE you entry.

Malaria is one concern, not so common anymore in Australia or Singapore, but the minimal risk remain. However, urban areas are safe.