When should I consider changing tyres due to age? The tyres in question are five/six years old Continentals.
You can tell a tire is ready to be changed when the rubber is brittle or when you're losing tread. I've found that tires that haven't been used or many years can sometimes get cracked and the rubber loses its flexibility, particularly in the sidewalls. If a tire is decades old, still has tread, and the rubber feels flexible, there's no reason to not use it.
When you can see cracks in the sidewall it's time to change a high pressure (over 75 psi or so) tire. Lower pressure tires are OK with a few cracks and likely develops cracks sooner anyway.
Also change a tire when the tread is worn down such that, with the tire partially deflated, you can feel a soft spot in the center of the tread. When the tire turns into a "slick" it's time to get rid of it. (Rear tires generally wear several times faster than front tires.)
Sunlight and ozone are hard on the rubber in a tire, so if a bike is stored outside (even if under a shelter) the tires will need replacement sooner. And tires with gum rubber sidewalls will develop cracks sooner than those entirely of black rubber. Stored indoors (or in a garage or outbuilding) black rubber tires can last 5-10 years.
And of course the sure-fire way to wreck a tire is to run it underinflated.
Cracked, stiff, brittle, or containing small puncture holes: if your older tires fit any of these four aspects, it's time to swap them out.
However, I would also ask yourself if you remember what those tires have been through. The tires may still be flexible and appear uncracked but be full of micro-cracks and punctures that you can't see with the naked eye. These small fissures can be disastrous when you're in the middle of a run. Spraying an inflated tire with soap & water can help find a leak, but many soaps will actually damage the rubber, making that technique counter-productive.
Unless you're just planning to use the bike for a quick run to the corner store, remember the old adage: "when in doubt, swap it out". If I obtain a "used" bike, or rescue one being tossed, replacing the tires and the tubes are the first things I do.