First of all, a disclaimer: I'm a noob, so take this whole post with a ginormous grain of salt.
Secondly, after finally finishing this novel: Egad, I'm long-winded.
Okay. You did mention it would be situational. There's a whole lot to think about. Will this city grow up and be a production city, or run commerce, or be a specialist city? The way you described it kind of sounds like a backfill city that won't be anything particularly special, that keeps things kind of basic. Is it on a potentially dangerous border? (I'm guessing no, because you said forget about military buildings.) Is it on a peaceful but cultural-intense border? Is there a resource you want that's three tiles away? Take into consideration your civ's traits and UB, they'll probably be important in the first few buildings. What civics you are running is another thing, you can get a lot of cheap buildings knocked out very quickly if you're running Slavery with a couple decent food tiles.
One more thing to consider before I try and give you an actual answer. You said mid-to-late game- depending on how late we're talking, all of those early Wonders are probably obsolete. So are some buildings- I have to mention Castles, even though you said not to, they're so much more than strictly for defense (I don't think there's any other building in the game that gives you so many different types of bonuses). Once again, though, they're probably unavailable by now.
Well, after writing all that mess, I don't really have a specific build order. Every one of my cities would be different. However, I'll tell you what I'd be thinking about, and why.
Okay. The things I'd have in place beforehand, so my infant city doesn't have to worry about them. Let's assume this is a planned-out city that's nice and safe, not some desperate culture grab on the border. Provided my empire has the available ability (a production city nearby with a few turns to spare would be nice), my target tile already has at least one defensive unit and a missionary either sitting there waiting or on the way. If I can spare a Worker or two, I'm already roadspamming the area and maybe pre-chopping some forests, ready to hook up resources and farm/cottage/mine/whatever as soon as the culture pops. The seafood resource already has a work boat waiting, and if there's annoying but "peaceful" neighbors nearby, I have a defensive Spy parked there too.
Ugh. I'm getting verbose. I said all that to say this: Anything your empire can do for your city, have it already done it advance to give your city a head start.
As far as the buildings themselves- if everything is nice and boring and not-at-all-complicated, my first two are usually Granary-Forge. Lighthouse and at least one culture building usually come soon after. But there are
soooooo many things to take into account. Here's a few:
City specialization almost always dictates what I'll build after the first few buildings, sometimes right from the start. However, the city you described probably won't be specialized, at least not to a great extent. If the tiles are average, and I'm not hurting for producting or commerce cities, I wouldn't be worrying about this too much for a little unimportant city. It's good to think about in the long run anyway, but it probably wouldn't affect my early build order.
Culture: How far and how fast does this city's culture need to expand? Obviously, the borders need to pop at least once to get the whole BFC. The missionary will do this in ten turns all on its own. If Stonehenge hasn't obsoleted it'll only take five.
But... if I need to pop borders a second time, that'll change up my build queue quite a bit. If you're running Caste System, you can have an Artist or two do this for you; or you can bump up the culture slider. I hardly ever do either of these, but they're options. If I need that resource sitting three tiles away, or if this city is snuggled up next to another civ's culture, one of my first buildings will be a Theatre, very cheap 3

PT. (Sometimes I build Theatres even before Granaries if I need to establish culture immediately.) If it needs another little kick, it gets a Library (which
almost all of my cities eventually get anyway), but they can take a while on a brand-new city.
Civics: Certain Civics can affect how I build things. If I'm in Nationhood (rarely), then Barracks gives me that +2

(if happiness is even an issue in a city this small). If I'm running Slavery (often), then I can whip-rush a
lot of smaller buildings very quickly, especially with two or three good food-producing tiles. Chop a Granary quick, then maybe Lighthouse if seafood is involved, then usually Forge. All the other early buildings will come very quickly.
Note: Lighthouses are usually not one of my first two or three buildings if I'm
not whipping, unless we're talking multiple seafood. That extra 1

from just one Fish doesn't seem to speed things along that much once the pop gets to 3 or 4.
But, if I
am whipping, a Lighthouse can really help when I'm trying to grow the city back up to 2 pop five times.
Oh, and about the Monasteries: Well... I'm usually in OR for much of the game, so I never build Monasteries for their intended purpose. I'll sometimes build them in a Science city after Library/University is done (hey, a lot + 10% = even more), or in a hotly-contested border city that
really needs the culture.
EDIT: While I was writing this, I totally forgot about the +2

from AP. I'm still not sure if Monasteries woud be in the first few buildings, but those hammers can come in handy. And that's another thing that can bump Monasteries (and Temples) up the list, too- if I happen to have AP/Sankore/Sistine/Spiral Minaret, those religious buildings really do a lot. If I don't, then they will usually wait for quite a while.
Leader Traits: Well, obviously, almost every trait has quickie buildings that go with it; basically, if it can get done twice as fast I'll probably prioritize it more, as long as it fits my strategy. If I'm Industrious (and I often am) the Forges get bumped even closer to the front of the list than they already are because once you have it every other build happens faster.
Unique Buildings: Playing as certain civs can have a big impact on what I build when. A few examples: Playing as the Aztecs (running Slavery, of course) a couple games ago I was building Granary/Sacrificial Altar in almost every city. It just makes the whiprush that much easier. Khmer's Baray helps the city grow faster, HRE's Rathaus is massive (especially if the city is far from the capital in a big empire), etc. Lots of civs have UBs that bump happiness, or health, or culture- if I need that bump, then I'm more likely to prioritize the UB.
Other various factors also come into play. I mentioned the Rathaus- even a regular Courthouse (which does take some time early on, unless I'm Organized) can be very high priority. If my economy is already teetering and this city pushes me too far into the red, I might make the city wait it out while building a Courthouse for the good of the nation. If maintenance isn't an issue, I might build ten other buildings first. If my happy and/or health caps are pretty high (yay resources!) then Temples, Colosseums, and Aqueducts wait indefinetely. Plus, if I have the right resources, Forges/Grocers/Markets do a better job anyway. You mentioned GLH/Colossus- if those two are still working, I try to get Harbors quickly, especially in a commerce city. As for the Cathedrals, save those rarities (can only make one for every 3-4 cities) for cities that have happiness problems and/or need massive culture.
tl;dr: Every city is different, so every city's build order will be different. Think about what you want this city to do and what its needs are, look at what each building will provide for the city, and go from there.
...
If this was way too long, and/or useless, and/or didn't really answer your question, apologies. I kinda got carried away.
