My bike does not have a UCI badge. Am I allowed to race it in UCI races?

Straight forward answer: It depends. Yes, you are if the frame was designed before Jan 01st 2011. If the bike does not carry a badge, it is possible that it will be examined by the commissairs. No big deal. Older frames are still ridden in world championships and do not carry the badge. This is perfectly fine. Forcing the riders to use frames with badges would hurt less priviledged countries more than others. This is certainly not intended by the UCI. Newer bikes without badges are not allowed in the races.

What is my source for the information: UCI frame approval protocoll. And verified at the Track Worlds 2021, where a bike frame was used that for sure has no badge. I talked to the owner of the company making the frame and our club owns one of the frames. This is why I researched. The written documentation is relatively sparse.

What about wheels, saddles, bars etc.?

Everything that you attach to your frame and is not wheels, needs to have a UCI approval. Period. It needs to be commercially available etc…

Wheels have to pass a homologization, if used in a mass start event. If you ride alone, you can ride the wheel you like. In the bunch, there are limits. Unless your wheels are standard as defined in 1.3.018 and do not feature rims higher than 2.5 cm, spokes thinner than 2.4 mm or less than 16 spokes. If not standard, your wheels have to be listed as approved. Check the UCI Equipment web page.