Okay, so I've looked into this a bit more.
It appears that when SVN times out, it's fetched an incomplete copy of the FPC source code.
I tried manually checking out the repos using TortoiseSVN (yuck, it forces integration into Windows Explorer. I am definitely removing this as soon as I no longer need it). I notice that it'll work, usually for a minute or two (in the best case it worked for 5-6 mins) before the transfer rate will drop to 0 and no progress will occur.
By cancelling this, (if nesseccary) doing a clean with "break write locks" enabled, and then trying to update the repo from the remote (<--- may be git-specific terminology here, I'm not familiar with SVN), repeatedly, I can eventually get the whole repo downloaded. I've gotten this to the point I have the full FPC source now, and FpcUpDeluxe will properly build FPC (before running into the same timeout error while setting up Lazarus). If I continue this, I should eventually be able to download them.
Unrealistic suggestion: According to Google, SVN is known to have these kind of issues in general for some people. FPC and Lazarus should move to a better source control, like Git.
More realistic suggestion: Lazarus and FPC do already have a GitHub mirror. If the devs can be convinced to actually put the proper release tags on that repo (or maybe they are there and I just overlooked them), FpcUpDeluxe just needs to support getting specific branches / tags from Git. Maybe it already does even, and it just needs options for the specific versions.
Even more realistic suggestion: Make FpcUpDeluxe detect the "timeout error" message and just try checkout again if it happens. Perhaps give up if it's happening repeatedly with no progress being made.
Best suggestion but completely outside this community's control: SVN need to fix their shit!