Shirou is the Red Dragon Emperor and Illya is the White Dragon Emperor.

Albion helped Irisviel make a direct connection to Illya from the Grail with out all the worlds evil corrupting her connection.

The Supernatural world of Fate Stay Night is a lie made by the true supernatural world of Highschool dxd, because of all the insane people in the Moonlit world. Heaven modified the Throne of heroes to allow the servants to interact with eachother as well as hide their knowledge of how the supernatural world really is.

Kuroka lives with Shirou hiding as a cat, waiting for a time to seduce him.

Kuroka and Koneko have made up due to a detective in the supernatural world clearing Kuroka's name.

Illya summons Fran instead of Heracles.
Rin Summons Atalante instead of EMIYA, because Counter Guardians and Alaya don't actually exist.(Artemis shows up and is overjoyed to see Atalante again, and tells her that she no longer is bound by chastity)
And Elizabeth Bathory was summoned instead of Cu, and she has a love at first sight crush on Shirou.
Medea is still struggling to find a master and finds Shirou.
Shirou still summoned Artoria.. but also Mordred and Morgana(saber) due to Ddraigs power.

Irisviel summons herself and removes the restrictions on the heroic spirits of the war so they are actually powered by the actual overflowing mana in the air. She also summons some more servants, being Kiyohime and Tamamo no mae.
And Illya has a crush on her little brother.

Albion gave Illya a dragon heart to replace the her Lesser grail heart, so she can have a long and healthy life.

Artoria and Mordred were previous red dragon and white dragon Emperors respectively, and still retain them as heroic spirits, artoria never used it, only trading her magic core, while mordred trained with it, but didn't use it against artoria.

After the Story, or whats left of, Stay Night is finished, they enter DxD story and later save Asia from the Fallen Angels and so on.

Manga! Issei. He's a much more morally right guy not entirely driven by lust or his hormones.