1.17 update on this one —
The phrasing on this 1st UHV may have changed a bit: it's now 4 Wonders in your capital in 1550. (It doesn't appear to matter if the Wonders are obsolete.)
From a 600 AD start, Theodosian Walls and Hagia Sophia are in Constantinople from the start date, so you should find them already present when you conquer the city. (This should be done on your first turn so that it becomes your capital — otherwise, a city that flips will become capital, and it won't be able to make enough Wonders in time. Great Bombards take care of the conquest no problem.)
Arabia and other civs may have left some Islamic or other Wonders that are accessible to you. However, the ones you will definitely be able to get (and are historical) are: Topkapi Palace (Judiciary); and Blue Mosque (Academia). Both of these Wonders have bonuses that are relevant to your special power/goals, so you will likely want to build them anyway.
However, a couple of things about this process surprised me, and made it less of a cake-walk than I expected.
Firstly, to ensure you get to Academia for Blue Mosque in good time, you will have to pay some attention to your science and economy, rather than just focusing on military and culture with a view to UHVs 2&3.
It may be best not to expand too much at first, to stop your tech costs from ballooning. I took Belgrade and Athens in addition to Constantinople, plus the two Anatolian cities that flip to you, but stopped there for a century or so. When you do expand further, head for more profitable areas like Syria, Mesopotamia, Egypt (which have more developed cities with Wonders) and leave the Black Sea and Vienna for later.
Grow Konya and Trabzon as much as possible given the terrain they're on (Windmills over Mines) as they're in your core. Other than necessities, all cities should focus on economic buildings for a while. Don't burn Population on your special Ottoman Draft power while it's only getting you Hvy Swordsmen (you start with a few Janissaries, but not the tech to build more), and you can also wait until you build the Wonder that limits the Unhappiness effect from it (Blue Mosque).
Doing all this will help keep beaker production up and tech costs down, but in order to reach your targets in time you may also need to pop a Great Person or two (see point 3 below), as few people will want to trade techs with you.
Probably, you will see Mongolia take control of Mesopotamia and Syria (hopefully they won't attack you!), before collapsing during the 1400s. This is a good time to take over the region, before Iran appears and does the same: better to fight some Crossbow garrisons than their Qizilbashi!
Secondly, Constantinople will have to get to Refined culture level (over 1000) in order to be allowed to build a 4th Great Wonder. Building some basic culture buildings (especially Theatre, Weaver) and running an Artist or two should take care of this in good time without major problems, but it's important to be aware of when you are planning your build order.
Thirdly, there may not be time for you to hard-build the Blue Mosque after researching Academia. This means some planning of Great People is in order. When you take Constantinople it will have two Wonders that produce Great Engineer points. Along with that, run one Great Engineer and no other specialists to make sure your first Great Person is a GE. (Mount Athos may help accelerate things too if it's in a city you take over.) Keep the GE in reserve until you have the tech and culture level to build the Blue Mosque, then rush it. (Hard-build Topkapi Palace: there's plenty of time for it if you beeline Judiciary first.)
Once you have your GE, you can run more specialists in Constantinople. I recommend Artists and Scientists, as either a Great Artist or Scientist can be popped to get towards the tech goal. Science output also helps with this, while the Artists help towards the final UHV.
After the Blue Mosque is sorted, your tech focus can change to Firearms so you can produce more Janissaries. After that, there may not be any more essential techs. (Your Unique Units are so good that developing better ones is not that important.)
A side-note on Civics: the Ottomans start with Despotism and Slavery active. These are out-of-date and not that useful, so I recommend switching to Monarchy and Manorialism for free on the first turn after capturing Constantinople (and any Worker hiding there!).
Out-of-date civics hurt your stability. With my limited early expansion plan, I didn't find that there were many opportunities to capture Workers, so cheap production of them from Manorialism was more useful, as was the Farm bonus (great combo with the Vassalage Farm bonus as always during this period).
As for Despotism, most of your cities are Production-rich and relatively Food-poor (except Constantinople, but you definitely don't want to burn your Population there), so whipping for production is not that useful. (If you do want to burn Population, you can always Draft.) On the other hand, Monarchy may be vital to keeping Constantinople happy for quite a while, until you have more buildings and can secure more luxuries.
However, so far I have only got through UHV-1, so I can't say for sure how all this leads into the subsequent goals!