Your fillings are either gold / palladium inlays or silver-mercury amalgam. The composite or enamel fillings don't apply here.
Welding fillings together will require that you have fillings in opposing teeth. It is certainly possible to weld metals together with a brief electrical current - it's done by robots in car factories every day. However, it requires quite a bit of current - without doing the math I would guess on the order of hundreds of amperes.
Your head (and, indeed, the tooth with the filling) is not a particularly good conductor of electricity, so the base current needed to weld your teeth together will require a much larger current flowing through your jaw. All those soft, wet squishy bits around your teeth will take some current too, reducing the available welding current and in general heating things up.
Your brain is surrounded by a saline liquid, and it's in a very well sealed solid enclosure (your skull). If we apply enough current across your skull that we have 50-100 amps across two fillings (and remember that this has to happen in the timeline of a lightning strike) we will also see considerable current and resultant heat around your brain. Water + enough heat = steam, and your skull is not a pressure vessel.
So, if your teeth do get welded together after a lightning strike it's really of no concern, because your head will explode.