You might have to go into the PS2 options screen (boot time, before the game starts - start the PS2 without a disc) and set the component video mode. There are two choices, RGB and .... well, I forget how the other choice is labeled. You want the one that's NOT RGB.
Also, just to be sure, I know you said component cables are being used, but many people confuse the terms component and composite. Composite is the one that comes with the PS2, component is one you have to go out and buy, has 5 RCA plugs, three for video, two for audio. The composite set only has three connections, one for video and two for audio. Just making sure; I know you said component and you probably know what it is, but some folks don't.
In the game options menu, when you select HDTV, it should switch temporarily to 1080i by itself to show you it can. Your set should switch automatically with it. Is it blank at that point, then comes back after the 15 seconds? Be aware that the HD display is not universal throughout the game. Menus, garage, etc. are plain ol' 480i, and it switches to 1080i (or 480p if that's selected) for the actual racing. It stays in the higher selection throughout a championship series if you're in one of those. Nor does it save the HD screen selection, you have to set it every time you start. (That keeps you from forgetting about it, taking your PS2 to a friend's house, and getting no use from it on his standard TV.)
Next, the point at which you throw your HDTV set off the nearest balcony (or put it up on eBay), you may have a set which will not display 1080i on component, but only on HDMI or DVI inputs. I've heard people say they had such, but I couldn't say for sure why an HD set wouldn't show HD on any capable input. If you actually HAVE to switch modes manually (and I can't imagine that to be the case) you're not going to enjoy this at all.
Also, look for this question on the GT4 forum, it's covered at great length there, and this is the same graphics engine in TT.
Lastly, and should have been firstly, you have the North American game and a North American console, right? Also, does the game display 480p if you select that instead?
Edit - I started THREE paragraphs with the word "lastly!" Fixed it, but how did I get out of school 7th in my class? Well, it was a long time ago. . . . .