As long as the processor is capable of handling the same arch, and you didn't compile a custom kernel that is missing anything, then yes it should all work perfectly. Just to clarify, if you are running a x86_64 system and the new processor isn't capable of 64-bit then yes it will break. Does not apply for working in the other direction, all 64-bit capable processors can handle x86 kernel and apps. Having said so, if you are downgrading and are running a custom kernel with the processor type set to i686, but the downgraded processor can only handle i386... unbootable system. However you are upgrading so you shouldn't have to worry about that.
Having said so, if you are just using a kernel from the repos then you are alright. Ubuntu compiles all of their kernels with compatibility for i386 processors, and with the majority of all distributed kernel modules included. This is to ensure maximum compatibility. In other words, yes it will magically figure it out, lol. Mainly because a basic ubuntu system has compatibility with most hardware on the market, you just only use what the kernel detects at boot or while running.