Let's change fate (pun totally intended) and see how the 5th Holy Grail War would fare with a mixed-up set of Masters and Servants. Let us take the standard set and then mix them up with the next best thing and see how that would affect the entirety of the Holy Grail War. This is Fate/Stay Night: Alternative edition.
First, a rule. Gilgamesh and Kotomine will remain a pair because they aren't a Master and Servant from the 5th Holy Grail War. They are more like interlopers from the previous war than an actual legitimate pair of participants. Otherwise, everything goes!
9 Saber And Rin
What would happen if Rin actually managed to summon Saber? The girl would be in a very good position in the 5th Grail war with Saber at her side.
Saber would be in a good position too. Unlike with Shirou, she would be blessed with an excellent mage that wouldn't hold her back. Her stats would also increase, leaving her a far stronger Servant in the war.
8 Archer And Shirou
Shirou is the catalyst in this case. If Shirou were to summon his moody but better self things would take quite an interesting turn.
If they were able to work together that is. Essentially Archer could very well turn Shirou into a less experienced version of himself. Together they could be quite the threat to the King of Heroes if they were both to spam Unlimited Blade Works against his Gate of Babylon. Of course, this is all considering that Archer doesn't Kill Shirou from the get-go.
7 Sakura And Berserker
It wouldn't be the first time that the Matou family would summon a Berserker Servant. In this situation, Berserker would still be passed on to Shinji to take Sakura's place in the Holy Grail War. Then he would bungle it up as he always does.
Shinji is a terrible Mage, terrible Master, and a terrible person all around. He is likely to send Berserker to an early grave or get himself killed. Especially with Rin and a powered-up Saber in the mix, there's a good chance that Saber might be able to kill Berserker a few times over with a single swing.
6 Atrium And Rider
Atrium Galliasta was originally the Master of Caster and then he got on her bad side and was killed off before the war started. He really shouldn't have insulted and belittled someone called the Witch of Betrayal.
Rider would probably end up in a similar situation. Part of the reason he treated Caster badly was that she has a history as a Greek villain. As it turns out, so does Rider. Her legend as Medusa would sow the seeds of mistrust and would eventually lead to him sealing off her power just like he did with Caster. In the end, Rider would either die or she would take him with her.
5 Bazett And Caster
Bazett Fraga McRemitz is the original Master of Lancer but was killed by Kotomine before the war started.
Bazett and Caster can be quite a formidable team. Caster is more than willing to play nice if she is given respect in turn, to the point where she considers the protagonist of Fate/Grand Order like a younger sibling if they get their bond level up. A kind and caring individual with a strong sense of justice like Bazett could end up in a similar position. Then they would be scary together. Bazett is one of the few magi that can keep up with a Servant and actually beats Saber in one timeline. If Caster were to support Bazett the same way she supported Kuzuki, she would be a threat.
At least she would if she didn't meet Kotomine and Gilgamesh first. Unfortunately, Caster and Bazett wouldn't be a match and Caster would probably be forced into a similar position that Lancer was forced into. Thankfully, she could then use Rule Breaker to get away from them and subsequentially would probably stumble onto Kuzuki all over again. They would once again join forces but with an extra focus on getting revenge on Kotomine and Gilgamesh.
4 True Assassin And Medea
There's a reason that Medea's Noble Phantasm is called Rule Breaker and it is because it does exactly that. Break rules. This allows a lot of fun manipulation, such as cutting off the contract of another Servant to their Master to even summoning one herself.
True Assassin would probably be put to the same job as Assassin did in Fate/Stay Night but would not do as great a job. While Kojiro was a duelist, Assassin is exactly that. An Assassin. He wouldn't fare too well in a face-to-face confrontation with a Servant and would work more like a speed bump to the attacking Servant and an early warning system for Caster.
3 Assassin and Zouken
What type of relationship would Zouken and Sasaki Kojiro have? Well, they wouldn't have one at all.
Zouken would order Assassin to do something and Assassin would have no choice by to do it as his tool. Zouken would use him to further his own schemes and would use his Command Seals and other diabolical means to get him to do so.
2 Lancer And Illya
Time to pair off the spares. Though considering that Cu Chulainn could be considering the Irish version of Hercules, it's not impossible to think that she might stumble onto this Servant while summoning. After that, they would be an excellent team in contention for the Grail.
Simply put, Illya is a great Master. She was bred and brought up to be the best type of mage a Servant could ask for and would be able to bring out the full potential of Cu Chulainn. With this combination, there would be a lot more Gáe Bolgs flying around in battle.
1 The Winners: Archer And Shirou (Though With A Caveat)
The caveat is that they would have to work together. If Shirou survives meeting Archer (maybe with the help of Rin's special gem) and Archer realizes that Gilgamesh is a bigger threat than his self-hate, they could win. Essentially, the Master would be a less powerful version of the Servant.
Then if Shirou would team up with Rin again, they could use her and Saber's support to survive most of the Grail War. Naturally, the team-up would also mean that Shirou has the potential to unlock his version of Unlimited Blade Works. Then they can do what no other Servant in the war can, take down Gilgamesh.
Medea and the Assassins wouldn't stand a chance. Rider is out of the war due to her contention with her Master. Lancer and Berserker could provide a fight but his divinity makes him vulnerable to Gilgamesh's Chains of Heaven. Even Saber wouldn't stand a chance without Shirou and Avalon to help her tank past Gilgamesh's immense power. Leaving only Shirou and Archer to contend with him. That being the case, if Shirou by himself could beat Gilgamesh then their combined might should do the same.