And to add further to the absolutely miniscule risk, there were 67.5 million people in the UK in 2022, and of those, 8700 tested positive for syphilis.
So there's that.
You've already figured out that the chances of this being syphilis is infinitesimally low, and you've had no other symptoms. You could easily go get another test to confirm this instead of just sitting and worrying, right? It will absolutely be negative.
Syphilis sores appear at the point of infection. It would appear on your tongue, not on the sides of your mouth. If your wife got a sore, it would not be a cold sore - that's oral herpes.
So test again and put this to rest and move on with your life. Don't go outside your relationship again - no judgement - clearly, your mental health can't handle it, and lesson learned. :)
She gave you VERY brief oral sex. Oral sex is already low risk for STD's and 10 seconds of it is really not time for impact. You also gave her oral very briefly and anilingus. You don't mention how long but guessing brief. If you had any risk at all, it was minimal at best. And your synopsis of her sore throat and your activity and your sore throat means most likely, you just have a simple throat virus and it was given to you from her but not an std involved. Just a regular virus. That we ALL get. You did kiss which transmits cold viruses. I do agree your anxiety is playing a role here.