Only the BTC node works normally. When the internet comes back, it will update the blockchain and start working again. It can even generate transactions without internet on the node, but they will only be advertisements when there is internet.
Discussion
Installation of bitcoin core and direct connection to sparrow, thats it?
I don't know how Sparrow works. I've never used it, but in theory if it has access to the BTC node (being on the same network) it should work. But the transactions created will only be advertisements and will enter the mining queue when the node has internet.