Some new information regarding Halo 3 content cuts has come to my attention and for the sake of keeping everything straight I've decided to write out what I believe the original storyline may have been.
As a precursor to this, here is some readily available source material:
everythingtranscript.fandom.com
www.halopedia.org
halo.fandom.com
So I've done some digging and have put together as best as I can, the original storyline for Halo 3 accounting for known content cuts... The game plays out the same through Floodgate but then as you reach the crashed flood ships, Cortana speaks to you directly rather than it simply being a message recieved onboard Rtas 'Vadum's ship. It would likely play out as follows
Cortana: "Chief! High Charity, the Prophets' Holy City, is on its way to Earth. With an army of Flood. I can't tell you everything. It's not safe. The Gravemind...it knows I'm in the system."
Chief: "Miranda, are you getting this?"
Cortana: ""But it doesn't know about the Portal, where it leads. On the other side, there's a solution. A way to stop the Flood, without firing the remaining Halo rings-"
Miranda: "We're on it, we'll rally the troops, you just focus on that ship.
The mission Cortana then plays out buuuutttttt you're actually on the crashed ship rather than high charity.
(As Cortana states in game 'High Charity is on its way to Earth..... if it did happen in game, its then to be assumed they immediately went through the portal as we never see it at Earth)
Based upon map names and the Bungie development video, an extended version of the The Ark is now played. Hoping to find a solution to the impending flood invasion Chief learns of the Guardians (my best guess being Guilty Spark reveils this information at the end of the mission infront of that map) Chief and Arbiter now spilt up, Arbiter wishing to stay near the Prophet.
And so the mission Guardian Forest begins (see links above) where chief collects the eye of a Guardian to be used as a weapon against high charity. The station otherwise being far too powerful for an all out assault by even Rtas 'Vadum and the Brute's fleets combined.
I don't have any information regarding the third mission known to be missing (090_ ?) I would guess it is at this point the scarab fight against the Gravemind takes place as it likely wouldn't take place in the Guardian Forest. Having destroyed the gravemind near/on Earth, the dynamic duo (Arby and the Chief) goes to give Truth two black eyes.
The mission 'The Covenant' would play out the same although without High Charity's appearance. The flood would have followed them through the portal and a 1v1v1 situation would occur during points in the the mission. Rtas 'Vadum's fleet would be indisposed glassing High Charity and it's (HC's) fleets, leaving the mission to play out the same as it did otherwise.
Mission 110_Cortana.... According to the in-game leadup, books, and some concept art I am guessing that Cortana turns... Having reached The Lost City, Cortana goes full on Halo 5 on us. Chief flees the lost city fighting against her 'sentinels and friends'.
Chief, unable to get Cortana under control would then go on to attempt to destroy Cortana and her new found weapons. In the mission Halo giving meaning to the Cortana moment quotes,
"Could you sacrifice me to complete your mission? Could you watch me die?
"There will be a great deal of hardship on the road ahead"
Faced with low ranking flood upon The Ark, and a rampant Cortana, Chief goes to light the ring with intent to destroy The Ark and eliminate the flood. Spark stops them.
Johnson: "We don't have a few more days!"
This time referencing the an increasingly powerful Cortana
And the rest of the game plays out the same as release, minus Johnson's final word's towards Cortana and her involvement during the Warthog Run.... She'd likely still have been present, just in a different way.
As a precursor to this, here is some readily available source material:

Halo 3
There are 10 missions: Arrival (Halo 3) Sierra 117 (Halo 3) Crow's Nest (Halo 3) Tsavo Highway (Halo 3) The Storm (Halo 3) Floodgate (Halo 3) The Ark (Halo 3) The Covenant (Halo 3) Cortana (Halo 3) Halo (Halo 3)

Forerunner City - Campaign level - Halopedia, the Halo wiki
100_forecity was a level that was worked on during the development of Halo 3, but was ultimately cut from the game. The level was to follow on from 090_alpine, and was ultimately combined with 090_alpine in the final release of the game to form the...

Cortana Moments
Cortana Moments are times throughout the Halo 3 campaign when either Cortana or the Gravemind telepathically address John-117. During these Moments, the player's movements will slow drastically, as will their ability to turn and face another direction. A screen effect will alter the display of...
So I've done some digging and have put together as best as I can, the original storyline for Halo 3 accounting for known content cuts... The game plays out the same through Floodgate but then as you reach the crashed flood ships, Cortana speaks to you directly rather than it simply being a message recieved onboard Rtas 'Vadum's ship. It would likely play out as follows
Cortana: "Chief! High Charity, the Prophets' Holy City, is on its way to Earth. With an army of Flood. I can't tell you everything. It's not safe. The Gravemind...it knows I'm in the system."
Chief: "Miranda, are you getting this?"
Cortana: ""But it doesn't know about the Portal, where it leads. On the other side, there's a solution. A way to stop the Flood, without firing the remaining Halo rings-"
Miranda: "We're on it, we'll rally the troops, you just focus on that ship.
The mission Cortana then plays out buuuutttttt you're actually on the crashed ship rather than high charity.
(As Cortana states in game 'High Charity is on its way to Earth..... if it did happen in game, its then to be assumed they immediately went through the portal as we never see it at Earth)
Based upon map names and the Bungie development video, an extended version of the The Ark is now played. Hoping to find a solution to the impending flood invasion Chief learns of the Guardians (my best guess being Guilty Spark reveils this information at the end of the mission infront of that map) Chief and Arbiter now spilt up, Arbiter wishing to stay near the Prophet.
And so the mission Guardian Forest begins (see links above) where chief collects the eye of a Guardian to be used as a weapon against high charity. The station otherwise being far too powerful for an all out assault by even Rtas 'Vadum and the Brute's fleets combined.
I don't have any information regarding the third mission known to be missing (090_ ?) I would guess it is at this point the scarab fight against the Gravemind takes place as it likely wouldn't take place in the Guardian Forest. Having destroyed the gravemind near/on Earth, the dynamic duo (Arby and the Chief) goes to give Truth two black eyes.
The mission 'The Covenant' would play out the same although without High Charity's appearance. The flood would have followed them through the portal and a 1v1v1 situation would occur during points in the the mission. Rtas 'Vadum's fleet would be indisposed glassing High Charity and it's (HC's) fleets, leaving the mission to play out the same as it did otherwise.
Mission 110_Cortana.... According to the in-game leadup, books, and some concept art I am guessing that Cortana turns... Having reached The Lost City, Cortana goes full on Halo 5 on us. Chief flees the lost city fighting against her 'sentinels and friends'.
Chief, unable to get Cortana under control would then go on to attempt to destroy Cortana and her new found weapons. In the mission Halo giving meaning to the Cortana moment quotes,
"Could you sacrifice me to complete your mission? Could you watch me die?
"There will be a great deal of hardship on the road ahead"
Faced with low ranking flood upon The Ark, and a rampant Cortana, Chief goes to light the ring with intent to destroy The Ark and eliminate the flood. Spark stops them.
Johnson: "We don't have a few more days!"
This time referencing the an increasingly powerful Cortana
And the rest of the game plays out the same as release, minus Johnson's final word's towards Cortana and her involvement during the Warthog Run.... She'd likely still have been present, just in a different way.