I was reading that there was disagreement on the block size limit and Bitcoin has been forked to Bitcoin XT? With two different flavors of the software running, which might result in two different longest blockchains, what is the impact on the current existing Bitcoin holders? Are they handled by Bitcoin Core or by both?
Bitcoin has not been forked yet. Once XT nodes mine 75% of all bitcoin blocks, XT will support a larger block size that will be incompatible with current bitcoin client. At that time, the blockchain is forked and the bitcoin currency would be divided into two, Bitcoin and BitcoinXT.
what is the impact on the current existing Bitcoin holders?
A transaction would be mined twice, XT miner puts this transaction on XT's fork; Core miner puts it on the core's fork. Then, you can run two clients, one of them spends Bitcoin, the other one spends BitcoinXT. Cool!
An SPV client always follows the one with largest proof-of-work, which means it can only spend coins from stronger chain.
Are they handled by Bitcoin Core or by both?
Both. You can double spend your coin now. One is bitcoin, the other one is bitcoinXT.
This fork means the devs on the new version of bitcoin (XT) would have complete control over any future updates on the bitcoin blockchain. As already stated, any coins held from before the fork would be able to be spent twice once Bitcoin-XT mine the first 8MB block. This is generally considered BAD, and is likely to drop the value of both bitcoin on both blockchains.