Wi-Fi in the saloons is relatively easy - just nail a few Access Points to the ceiling (incidentally, Wi-Fi is facilitated by "Access Points" not "Routers" - this isn't just hair splitting over nomenclature, in the field of data networking an "AP" and a "Router" are very different things - the "get-you-on-the-Internet-omni-box" you have at home contains both and a lot of other stuff besides.) 2-3 AP's per carriage would be my preference, depending on predicted pax. numbers and how good a service one wishes to avail to each Wi-Fi device. These would then all be lashed into a grand "on train" (wired) network and delivered to "something else" to upink to the rest of the world.
As others have indicated, the "hard" part is the uplink from the train to the rest of the world, especially since trains move - some of them very fast. Even with well known paths ('cos trains run on tracks) and clever antenna design and positioning, I'm skeptical that existing 802.11 (Wi-Fi) standard could provide the bandwidth and reliability to service a train load of pax. all wanting to watch Eastenders on their phone. With Wi-Fi's relatively low xmit power, a fast moving train would not stay "in range" for long before needing to "hand off" (roam) to the next base station.
And of course, all the Wi-Fi device inside the saloons and near by lineside premises with their own Wi-Fi, mutually interfere with each other and any lineside Wi-Fi devices facilitating track-train comms. (I've been know to run a "sniff" for Wi-Fi AP's on my West London to Central London commute and can find literally hundreds of AP's fighting for/over/polluting the airwaves over a mere 15 mile route.)
One could easily believe that no-one has trailed this as - even as a thought experiment - one can foresee so many problems using Wi-Fi for high bandwidth and reliable train-tracks comms.
With limited bandwidth available ad potentially large numbers of pax. to service, it would not suprise me if TOps curtail or inhibit streaming video and music. I tend to think "something else" would be used for the uplinks such as 3/4/5G others have discussed.
Interestingly, London Underground are using Wi-Fi for some of their modern signalling - IIRC the Jubilee line has "wiggly wires" down the tracks and the sub-surface lines are rolling out loads of Yagi antenna and so forth.