As the title suggests, in this world there are only three metals available in any meaningful amount: copper, gold and silver. How advanced could computers get in this world? Could things like the microchip or even integrated circuits develop?
The history of computers starts with mechanical devices. In early history, the Greeks had some mechanical computing devices. Many early devices were for figuring out phases of the moon or seasons. These could be made from copper alloys. https://www.britannica.com/technology/computer/The-first-computer
There were a number of mechanical computing devices used throughout the centuries including the slide rule. https://en.wikipedia.org/wiki/Analog_computer
Electronic computing started with vacuum tube and relay technology. This can be done with copper.
So, yes, computing can be done with just those metals.
As far as transistors and integrated circuits, those are based on silicon (not a metal) with various doping agents added to the silicon to change how it operates. https://en.wikipedia.org/wiki/Doping_(semiconductor) Gold is often used for wiring.
Now, the problem will be the rest of the world. On earth, aluminum is a common metal combined into various compounds. For example, clay is an aluminum compound (https://en.wikipedia.org/wiki/Clay). So, your planet can't have pottery. The ocean floor and mantle are mostly basalt which has a lot of magnesium and iron. The earth core is mostly iron and nickel. Your whole planet will be very different from earth and is likely not to have a magnetic field protecting it from solar wind.
Absolutely! But first a little history:
While today we use a specific style of silicon chips called "CMOS" (where all the 'switches' come in complementary pairs, one to pull up towards a "1" and one to pull down towards "0"), this was not always the case. Some of the earliest used something called Diode Logic which actually needed no transistors at all - why is this important you ask? Well...
Copper oxide can be used to make diodes and the only other thing diode logic needs is a resistor, which is literally just anything that can't conduct a lot of current so a fine metal wire (or a lump of charcoal) works just fine. Combine your copper oxide diodes and your coper wire resistors and bam! You now have digital logic. Now don't get me wrong, you can't build very complex, fast or efficient logic with this (we settled on CMOS for a reason) but it IS made only from copper (and oxygen) and it IS digital logic and what is a computer but a bunch of digital logic?
So, can you make a computer solely out of gold, silver and copper? Absolutely!