The matchmaker visiting times are 10:30 AM, 3:00 PM, and 7:00 PM. She will visit at these times for three days, beginning a couple of days after your Tamagotchi becomes an adult.
Pausing your Tamagotchi delays its development, so it may take a while longer for the matchmaker to show up, if you've paused your Tamagotchi.
If your Tamagotchi is old enough, you can do a clock-set cheat to get the matchmaker to appear: set the clock on your Tamagotchi to 10:29 AM, 2:59 PM, or 6:59 PM, and then wait one minute. If you set the clock to the time she's supposed to appear, 10:30 AM for example, she will not appear. She appears as the clock ticks over, from 10:29 AM to 10:30 AM for example.
If the matchmaker doesn't appear, your Tamagotchi may not be old enough. Wait until tomorrow.
If you miss the matchmaker, or refuse her offered mate every time she visits, your Tamagotchi character will turn into an oldie. The matchmaker will not visit oldies.