So you're saying I will only need to swap out the wallet.dat file and then restart the client without rescan?
Yes, it will actually perform the rescan itself if it needs to. You don't actually need to force a rescan.
I tried to switch out the wallet.dat file before downloading the blockchain, but since this is a fresh installation there was no new wallet.dat.
You can stop it now and swap out the wallet.dat file. It may actually be better to do that to avoid a longer rescan later. The sync will pick up where it left off.
Once the chain is synced, I simply swap the wallet.dat files, restart Bitcoin-qt normally, and my coins should be restored with no need for dumping the key or rescan?
A rescan is still required, but it will be started automatically by the software.
Will it have to rescan the entire chain (a week) all over again or will it take around 30 minutes or so?
Rescanning is not the same as downloading the blockchain or reindexing the blockchain. It will be much, much, much faster. On the order of tens of minutes rather than days.