You may be asking: “Why is it important that my program is accredited?” and “What is accreditation?”

According to APS, accreditation “refers to the process of external assessment to determine if a course (and the higher education provider offering the course) meet the Australian Psychology Accreditation Council (APAC) Accreditation Standards for Psychology Courses. Courses which do not achieve APAC accreditation are not eligible for Approval by the Psychology Board of Australia (PsyBA) as programs of study suitable for the purposes of registration as a psychologist in Australia.” This applies the same to the overseas courses provided by universities in America and UK, and hence will be accredited by the APA and BPS respectively.
In short, if the program you are doing is not accredited by either the APA/APS/BPS, your program will not be recognised to allow you to be registered as a psychologist. This explains for why taking accredited programs is important.

For those who wish to check whether their overseas courses are accredited, here are the details:
Accreditation by SPS:
SPS does not accredit any courses, but recognise any degrees that are being accredited by the respective accreditation councils by the societies/associations. To quote them, “The general rule is that the psychology degrees must be recognised or accredited by the national psychological bodies in that country, or are completed at and conferred by universities that are recognised or accredited by the relevant authorities in that country, or are recognised as qualifications to apply for registration or licensure as qualified psychologists in that country.”

Accreditation by APA:
APA-CoA only accredits “doctoral graduate programs in clinical, counseling and school psychology, and programs offering combinations of two or more of these areas. The primary professional degrees offered are the PhD (Doctor of Philosophy) and PsyD (Doctor of Psychology).” To add, “The APA-CoA does not accredit schools or universities, or programs at the bachelor’s or master’s levels.” To check on the individual programs, please check on the links to check respectively.

 Accreditation by APAC:
For APS,  which is the accreditation body for psychology in Australia, all programs have to be accredited, from all the undergraduate to the postgraduate programs. Only those programs offered by the universities in the list and conducted in Singapore are accredited by APAC. From there you can check whether your university and the program is on the list.

Accreditation by BPS:
In the BPS link, you will have to select your respective program (undergraduate/postgraduate), and go through the list to see if your program is in the list. You may also search by selecting the institution/university to see if the programs are available. Similar to APS, all programs have to be accredited by BPS.
(A post in the SPS facebook has mentioned: “The BPS will not give automatic accreditation to any program taught outside the UK. If you do a British degree at a Singaporean institution, you have to apply individually for GBR and BPS membership. Your British university should support and assist you with this application.”)

Just a note: Accreditation is different from the program being permitted to be conducted by the Council of Private Education (CPE) in Singapore. To check whether your program (conducted by a private education institution/foreign institute) is permitted, please check via this CPE search.
From my understanding, your overseas program is recognised in Singapore, as long as it is accredited by APA/APS/BPS, and also permitted by CPE.