There's actually fanart of this team and it's adorable. Honestly, without the big kaiju threat of Gilles' Noble Phantasm the two of them might actually win the war - if only by virtue of not really caring about it in favor of murdering muggles while the other six participants kill each other. If the last Servant standing is female, or if the last two Servants score a mutual kill, then that's it. Between Jack's presence concealment/information erasure wombo combo and their shared ignorance/apathy regarding the War, they probably won't be discovered until the very end, when either the Grail appears in front of them or the last man standing realizes that Assassin never showed up. And knowing Ryuunosuke, by that point he'd have connected with Jack enough that his wish would be for her to find her mommy instead of something horrific. Which would probably end with a lot of dead women.
...well, unless he realizes that they both get a wish. In which case, Jack probably gets to stick around and the body count will keep climbing forever. Angra Mainyu is pleased.
Jack and Ryuunosoke are all set to be a problem yep. Interesting points to consider: in canon Apocrypha Assassin had to feed indiscriminately because her Master was a literal muggle with no ability to supply her with prana. I think that as coming from a Magus bloodline Ryuunsoke could provide Jack with more energy, so they be acting somewhat more subtle than Giles canon rampage.
Actually between Kirei not having a army of Assassin's to act as city wide scouts, Ryuunosoke being a serial killer whose killings have already been reported and Jack not likely to think Saber is her long lost love Team Assassin could fly under the radar for a bit longer than canon. Of course Jack's out of her depth the moment someone actually fights her.
The only Servant she would have a real shot at taking down one on one is Archer due to her being a woman and her having a soft spot towards children.
Congratulations, Kayneth, you have won the war. All it cost you was your ego being torn to shreds and hung out to dry.
...is what I'd like to say but for my complete disbelief that there will ever be a single Grail War scenario where Karna appears and is not forced to job.
More seriously...
Starting with the obvious cases, Kiritsugu and Kayneth starts in pretty much the same circumstances as in canon, permitting the same initial strategies, and their Servants are not ones to show initiative and brow-beat their Masters onto a different course. Siegfried is capable of stalling Karna means that ultimately, their conflict will reach the same resolution - with KFC. That said, Sola-Ui will probably act more responsible, and Karna's loyalty has a bit different flavour from Diarmuid's (far less romantic notions, for one) so this team still has a shot.
All others can go too many ways for me to comment on without further consideration. Though, from a narrative standpoint, another thing that makes a lot of sense is Siegfried giving his heart to Shirou during the climactic fire, thus providing the set up for an eventual sequel, saddling Shirou with a hero complex of a slightly different nature, and giving him an alternate method of surviving some suicidal decisions.
Karna ends up being Kayneth and Sola's marriage counselor. He forces Kayneth to confront the fact that Sola resents the arranged nature of their relationship and then forces Sola to acknowledge that Kayneth does love her and would give up anything to keep her safe.
I did think of the Shirou gets a Servant heart thing but well another idea occurs. Siegfried canonically donates his heart to give a doomed artificial human a shot at life, what happens if he after figures out that Irisviel's days are numbered he decides to help her out by giving her his heart.
The effect of the transplant on Sieg in canon reinforced his body and allowed it to potentially survive far beyond its built in expiration date. Would such a transfer allow Iri to survive beyond the Grail's manifestation?
Because if that's true then well a lot of domino's fall differently, Kiritsugu is confronted by an act of selfless heroism he can't mock or find fault in, Iri is now bluff and can transform into a Servant. That could throw things just slightly out of whack.
Tokiomi definitely has little chance of making it to the end on account of being an opressor. Invincibility Trio are pretty much the frontrunners here; Jack is stymied by this being a sausagefest, Shakespeare doesn't have the firepower needed, and Atlanta's okay but not Invincibility Trio.
(This doesn't take into account varying situations in the inevitable cluster fuck)
Honestly I'm wondering if Kariya and Waver could team up in this scenario. Archer isn't a mindless killing machine and can actually offer her master advice on how to proceed. Given that along with her and Achilles's compatibility in canon it won't be a stretch for her to go
"Hey Master, Rider's master is a novice in over his head, who already has Lancer's Master out for his blood and is an outsider so the Tohsaka and Einzbern are going to want him gone yes?"
"Sounds about right, where are you going with this?"
"Someone like that has to be desperate for allies and we could use the extra firepower. Perhaps we should consider an alliance."
And just like that Waver & Kariya become a major power in the war.