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Many of the Church Fathers say that the chastisement precedes the period of peace, then after the peace is over, the time of the anti-christ comes.  The chastisement is NOT during the anti-christ's time, this is for sure.  Our Lady of Fatima does not talk about the anti-christ, but she does talk about a chastisement in the form of revolutions, martydoms and social upheavals (culminating in the 3 days of darkness, which is prophesized elsewhere).  The 3 days of darkness is NOT connected to the time of the anti-christ in any prophecy that i've ever read, so this is another sign that the time of the chastisement precedes the anti-christ.
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Some confusion arises when we hear of the "end times" and many people correlate this phrase to the time of the anti-christ only.  But many of the Church Fathers talk of these 3 time periods - the chastisement, the peace and the anti-christ - as the 'end times", because all together, they are connected.
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One thing that the Church Fathers are unanimous on, is that the Great Apostasy is not happening now because before the anti-christ comes, there are 6-7 different things that must happen to the world before his arrival.  One being that the whole world will be Catholic, which has not happened yet in all of history.  The USA has never been a catholic country.  Same with China, Japan, Thailand, etc.  The period of peace will be the time when the whole world will follow Christ and his Church like never before.  This is why when the anti-christ comes and seduces the world, it is called the "Great Apostasy" because all those who follow him into error actively reject catholicism.
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The Chastisement that we have seen since V2 and which may or may not culminate in persecutions, and possibly an outlawing of the mass for a time, is not the same as will happen under the anti-christ.  In our period, the trials we may face will be a precursor to those of the anti-christ, but not be as long or as encompassing.  In our period, Our Lady will triumph over satan's forces (i.e. Fɾҽҽmαsσɳɾყ), while in the future, Christ will triumph over the anti-christ and satan himself.  In our period, Our Lady will save those countries and people who are catholic and then convert the world as a result of Her triumph.  In the future, Christ will save all countries and peoples and the world will end.  In our period, the mass may be outlawed in certain countries but not all (since it's not said everywhere).  In the future, the anti-christ will outlaw the mass for the entire world.
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The best explanation for all of this is that the Chastisement is the end of the 5th age of the Church, with the 6th age being the period of peace and the Age of Mary, and this peace followed by the rise of the anti-christ in the 7th and final age of the Church.  This is Ven Holzhauzer's explanation and it agrees very well with many prophecies and with the volumes written by the Church Fathers.

If the whole world witnessed the Chastisement and converted to Catholicism, why would they ever apostasise? It's an issue that confuses me. Also, which Church Fathers spoke of the Great Chastisement and the Period of Peace? I was under the impression that we get most of what we know about those from the prophecies of Saints and Marian Apparitions. That's all I can find when I google, anyway. 

If the whole world witnessed the Chastisement and converted to Catholicism, why would they ever apostasise? It's an issue that confuses me.

Most of the world will die during the chastisement and those left will be catholics and those non-catholics who will mostly convert (the exception being some jews, since the jews will not convert en masse until after the death of the anti-christ).  Then the whole world will be catholic there will be peace.  However...just like we witnessed in our times the fall of catholics from the 40s/50s to the post-V2 time of the 70s, this shows us that it can only take 1 generation (or 30-40 years) for people to forget their faith.  The answer, simply, is that the generation born post-Chastisement will 1) not know enough about the evils of the pre-chastisement world and/or 2) will not appreciate the Faith/peace that they enjoy and thus 3) their Faith will not be as strong and they will be also be spiritually weakened by the "good life" of peace and worldly prosperity.  They will fall victim to the anti-christ's errors because human nature doesn't change and the triumph of Our Lady will not mean as much to them as it will to us.
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Another aspect to all of this is that the transition period between the "period of peace" and the anti-christ will probably be 15-20 years or so.  The end of the period of peace is the death of the Great Monarch.  When he dies, then the Holy Roman Empire is divided up into 10 kingdoms and this is where Scripture says that the 10 kingdoms fight against each other and the anti-christ gains world fame by winning battles at a young age.  But he does not gain full world power until the age of 30, which mocks Christ's age at His time of public life.  So if the anti-christ is a military prodigy at, let's say, the age of 10-15, and he doesn't gain world power until 30, that means that we have 15-20 years of a transition period.
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If the period of peace is short (as many prophecies say), let's assume that's 20-25 years.  Add 15-20 more years of transition between the peace and the full reign of the anti-christ and you have 35-45 years.  At the time of the reign of the anti-christ and the start of the 3.5 year persecution, anyone younger than 35-45 years old would have absolutely no memory of the passion of the Church or the start of the Great Monarch's reign or the Chastisement or life before that.  Anyone younger than 45-55 would've been 10 years old when the Great Monarch died and also have a vague and childish memory of the Church in Her glory.  So it's easy to see how this generation could be led astray by many of the same errors which existed in our times/pre-chastisement, because these people didn't see such errors affect their daily lives like we do today.  They grew up in peace; they grew up with the Faith and didn't have to work for it.
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Just like we see the young generations of Tradition today, anyone younger than 40 does not remember V2.  They hardly remember the papacy of JPII (and if they do remember him, they don't remember his shockingly scandalous early days when he was hereticlly promoting the one-world religion and other crap).  They don't remember +ABL or when the sspx started in the 80s.  They don't know about the time period pre-indult, when modernist rome didn't allow any latin masses from 1970-1986.  They don't understand why older Trads are anti-indult, why older trads have a problem with the new ordinations, why older Trads have a problem with the FSSP.  Only those younger trads who study history, pray and listen to their elders have kept the Faith and stayed on the course of Tradition. 
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Under 40 trads today don't see a problem with the indult, they don't see a problem with a "conservatively said" novus ordo mass, they don't see a problem with +Benedict's error of interpreting V2 "in light of tradition".  They are too young to remember the post-V2 war for the Faith.  They are too young to appreciate sacrifices made to keep Tradition alive.  They are too young to appreciate what they have been given.  ...So it will be at the time of the rise of the anti-christ.  Our times are a parallel, a precursor, an echo of the times to come.
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Also, which Church Fathers spoke of the Great Chastisement and the Period of Peace? I was under the impression that we get most of what we know about those from the prophecies of Saints and Marian Apparitions. That's all I can find when I google, anyway. 

I don't know specifically but a book I read which summarized their writings, says that most of the Church Fathers talk about a period of peace before the reign of the anti-christ and this period will be when the Holy Roman Empire will return for the 3rd time in History (which ties into the Great Monarch prophecies). 

Very informative posts, thank you. I'll be sure to read up more on it later.