It depends on the scope of your mod, but here's a rough idea:
1> Make a backup of the old code. Obviously it's too late for you to do this now.
You really only need to backup two things: the assets/Gameplay directory (which contains all of the gamedata XML, as well as a few key Lua functions like AssignStartingPlots) and the assets/UI directory (which contains all of the important Lua). Do not try to copy the Resources directory, because it's just huge.
To avoid this in the future, you can tell Steam to not auto-update Civ5. That way, you could wait until the last second to make the backup. But I wouldn't do that.
2> Go download a good comparison tool. I use WinMerge, but any emacs can do this pretty well.
From this point on, there will be three copies of every file:
A> The pre-patch version of the official code
B> The post-patch version of the official code
C> Your mod's version
Obviously, comparing A with B will tell you what changed.
3> For Lua and non-GameData XML files that you've modified from existing code (the things you set VFS=true), all you need to do is find the official file. See if the "last modified" date matches the new patch.
If not, then there's no need to do anything. Nothing has changed in the official code, so you're done.
If it's changed, then compare C (which will be A with some small changes) with B. In most cases, you'll simply be able to copy anything that changed from A to B directly into C. Only rarely will there be a direct conflict with your own changes.
(Because you're comparing C with B directly, without using A, you don't need the backup directory for this. But it's safer anyway.)
Note that non-GameData XML, in this context, includes things like the art defines files, which are stored inside the .fpk files and need to be unpacked through Nexus. A patch won't update the "loose" versions of these files, so you'll need to unpack them gain, and this might screw up the timestamps. So for those, you pretty much have to assume that they've changed, unless you copy over all of the loose files to your backup directory.
4> For custom-made Lua files, there's nothing to do. The patch added several new functions, so there might be ways to improve your code, but the old code will work just fine.
5> For GameData XML or SQL, you need to compare A with B. (C won't be a complete copy of a file, it'll only contain the new Rows and Updates.) Then, go through ALL of your own files (C) and see if your changes still make sense. There's no easy way to do this; in most cases a bad XML entry won't crash the game, but it will cause screwy things to happen.
Common issues include:
I> Duplication of text keys.
For instance, in this last patch they finally added TXT_KEY_YIELD_SCIENCE. This is important for things like if you made it so that Farms generate +1 science at Fertilizer instead of +1 food, or something like that. Before this patch, that text key didn't exist, but now it's in there. If you don't go back and remove any duplicates, you'll see TXT_KEY_WHATEVER everywhere, because any file containing a duplicate key is ignored entirely.
II> Multiplication of effects.
I'll give two examples from my own mod.
Public School: was originally +50% science. My mod used an Update to change it to +40% science instead. Then the patch changed it to +1 science per population. End result: the mod would try to change the now-zero science modifier to 40% (but would fail because there's no longer an entry in that table), and the +1/pop would be added, but the text key would say +40%.
Planned Economy: This policy originally gave -50% to unhappiness from number of cities (effectively +1 happy per city). I decided that it was stupid for a policy named "Planned Economy" to be about Happiness, so I changed it to "+20% income from trade routes" and set the unhappiness modifier to 0.
Yesterday's patch changed it to "+25% research in cities with a Factory", which makes even less sense for a planned economy. But if you tried running my mod with it, it'd add the 20% income, lower the unhappiness modifier to 0 (except it was already there), AND still get the +25% research. So now it would give both gold and research, but the help text would only mention the income.
III> General rebalancing.
You add a tech in the future era. Now, suddenly, those techs cost about 30% more than they did yesterday, so you have to go back and reprice your tech. Same goes for units; most units are now cheaper, especially in the later eras, because many of the +production buildings were reduced in effect (and the Arsenal's bonus is gone entirely), so if you added a new unit its price will need to be adjusted. Many air units had their strengths reduced, but the Weak Ranged promotion for fighters went from -75% to -50% so they'll be even stronger against ground units than before. Movement rates of units get tweaked.
Content-heavy mods have major issues with this. My own mod adds 48 techs, ~70 Wonders and Buildings, ~30 units, 10 Policies, a bunch of promotions, etc., all of which need to be re-evaluated for balance.
IV> Fundamental design issues.
This is the most painful. Rarely, the devs will make a change that so alters the underlying game systems that you now have to go back and see if your mod needs to be fundamentally restructured. Maybe some parts can now be removed entirely; you might have put in an anti-ICS tweak, but the devs have made a different tweak that already fixed the problem, and now you have to decide whether your change is even needed. (In my own mod, I'd lowered the research output of buildings, to slow down tech advancement in later eras. But in the last two content patches the devs have raised the costs of all late-game techs, so my reductions might not be needed any more, and might in fact go too far.)
Sometimes it's not even obvious at first. The devs add "Finisher" policies, but you've changed how the policy trees are structured to where those finishers now totally unbalance things. Not an easy issue to resolve.
6> Once you've done all of the above (generally takes a day or two for large mods), go play a game. See if anything stands out as being wrong; check the tech tree, check the civilopedia, check the tooltips for policies and units. Don't be surprised if you find screwy things happening; it'll take quite a while to iron out the small issues.
