This story originated and was first publicized by the Jewish Christian and long time Christian Missionary to the Jews, Ludwig Schneider, on page 18 in the March 2008 edition of Israel Today Magazine. Here is the link.
He states clearly that this is found in Sefer HaGematriyot of Rabbi Yehuda HaChassid (Yehudah ben Shmuel).
I have the sefer and to date have been unsuccessful in locating the citation he refers to. When I have questioned anyone, Jew or otherwise, as to the precise citation, no one seems to be able to produce an exact page or parsha. The book is long and the gematriot and other tzerufim are numerous. I will, bli neder, continue to go through the sefer and if and when I find the citation, will report back.
To complete the report on this citation, it does not exist in Sefer HaGematriot of Rabbi Yehuda HaChasid. So the report from Ludwig Schneider is false.
The actual source for this concept is from the Zohar, Volume 2, parshat VaEra, page 32a beginning with the words ות״ח, ארבע מאה שנין. It is a statement of Rabbi Yossi said in the name of Rabbi Shimon bar Yochai. It explains that the 400 years which are derived from the 8 Yovelot correspond to the reward due to Yishmael for the fact that he was circumcised by Avraham. The normal time for circumcision (Milah) is on the 8th day after birth. Since Yishmael was already 13 years old, he isn't credited with proper fulfillment.
Schneider says this is a calculation relating to a 400 year empire that would rule over Jerusalem. The Ottoman Turkish Empire, which was formed by Osman I in 1299, didn't exist at the time of Yehudah HaChassid's death in 1217. Osman I lived from 1258-1326. So whatever reference Rabbi Yehudah HaChassid might have made is not about the Turkish Empire. It would be generic. If the citation is accurate, the association to the Turkish Empire is speculation by Schneider.
For the time being, the time line stated by Schneider is the following:
He says, Allenby conquered Jerusalem from the Ottoman Turkish empire on the first day of Hanukkah, December 9, 1917 (24 Kislev 5678). He counts back 400 years which he says corresponds to 8 Yovelot (Jubilees). There is a Torah sourced basis for this 50 years per Yovel calculation. That places the year at 1517 (15 Kislev 5278) which corresponds with the approximate time frame that Selim I of the Ottoman Empire conquered the area of the Middle East which included Jerusalem from Shah Ismail of the Safavid Persian Empire.
The significance of the removal of control of Jerusalem from the Muslims, the Ottoman Turkish empire is based upon Yechezkel 46:17.
וכי־יתן מתנה מנחלתו לאחד מעבדיו והיתה לו עד־שנת הדרור ושבת לנשיא אך נחלתו בניו להם תהיה
"But if he gives a gift from His inheritance (meaning Yerushalayim, the undivided Capital of the land of Israel) to one of His servants, it shall be his until the year of liberty; then it shall return to the prince. His inheritance shall be only his sons'; it shall belong to them."
The 'Prince' (Nasi) being referred to here is an abbreviation of 'Nitzutz Shel Yaacov Avinu". It means the entire Jewish people as a single entity, the idea of 'Tzibbur'. And so Yovel (Shnat HaDror) starts to be counted either when the 'possession' begins to be removed from the servant (the Islamic Caliphate), which would be 1917 or when it actually comes into the possession of the the nation of Israel (the children), which would be 1967.
Schneider points out that from 1917 until 1967, another Yovel of 50 years, Jerusalem was like a "no mans land" with control shifting between Britain and Jordan. During that time period, the modern state of Israel was established on May 14, 1948 (5 Iyar 5708). And then, on June 7, 1967 (28 Iyar 5727), Jerusalem was reunited under the Jewish state's control during the Six Day War with Rabbi Shlomo Goren actually ascending the Temple Mount and blowing shofar from the Dome of the Rock.
To complete this picture, Christians also try to connect this with the blood moon prophecies found in Yoel 3:3-4, 4:1-15. These particular prophecies speak to days of judgement for the nations of the world. Yoel 3:3 in particular is classically associated with the beginning of the first of three wars that comprise the Wars of Gog u'Magog. (See Sefer Avkat Rochel by Rabbeinu Makir ben Abba Mori, Book 1, Signs 1 through 5. It seems to be associated with the detonation of the two Atomic bombs at Hiroshima and Nagasaki which began on August 6, 1945 (27 Av 5705). The "Date Palm trees of smoke" corresponding with the "Mushroom Clouds". The shape is the same.
As is known, following shortly on the heels of the reestablishment of the modern state of Israel, a blood moon cycle started on April 13, 1949 (14 Nisan 5709). This also happened again just before the reunification of Jerusalem on April 24, 1967 (14 Nisan 5727).
As is common knowledge, we just completed another blood moon cycle this last Sukkot, September 28, 2015 (15 Tishrei 5776).
It is worth pointing out that the conclusion of this section of Zohar quoted above says that following this period where the Bnai Yishmael merit to occupy the land of Israel during its desolation, they will precipitate three very harsh wars in the world. One is by Sea, and one is by Dry Land and the third is about Support of Jerusalem. And it says Israel will not be delivered into the hands of the Bnai Edom in these conflicts.
These first three conflicts appear to be associated with what are called the Gulf War (by Sea), August 1990-February 1991, and Desert Storm (by Dry Land), January 1991-February 1991. The third element of this conflict resulted from the attacks on Jerusalem by Saddam Hussein by firing Scud missiles. This emboldened the Palestinians and Hezbollah to continue rocket and missile type attacks against Israel. This third stage continues to this day.
The Zohar continues and says that during this time period, one nation from the ends of the earth will make a war against Rome (the capital of Edom) which will cause all the Bnai Edom from the whole world to come and fight them. It says this conflict is associated with the posuk from Isaiah 34:6 which says HaShem makes a sacrifice of Basra. It says at the conclusion of this conflict all of Bnai Yishmael will be removed from the land of Israel by HaShem.
This additional conflict coincides with the Afghanistan War (September 11, 2001 to the present) and the second Iraq War (March 2003 to the present). The Afghanistan War started with the attacks on both New York City, the financial capital of the United States and Washington DC, the political capital of the United States. The city of Basra in Iraq has been an ongoing center of conflict since the second Iraq War broke out through the influence of the Mahdi Army.