A 4 slab on the second level seems odd to me, but I am out in Oregon so our 1910 homes used more wood than rock.
So a few things, I am assuming that the walls are lath and plaster, it cracks easily and repairs are annoying and generally look like repairs. If you start jack hammering, the floor out you could cause cracks to the ceiling below, you could let a block of concrete loose and have it fall below destroying the floors / people / pets or furniture in its path. Unless there is plywood or shipboard under the concrete, that will help support the pieces and allow you to pick them up and remove them.
You have to decide where you want the project to end, if you only take up the tile you still have this concrete slab and old plumbing. Something I found very interesting is that many homes in the early 1900s did not actually have indoor plumbing, so it would have been retrofitted into the home after construction. This means that the location where pipes are run could be different than what you would find customary today.
Whenever I remodel a room, I remodel the whole thing, and go out as far as possible so I can tie into the new stuff when I do the next room. I gut out the floors, the walls, wiring, plumbing, insulation and start off fresh. Generally I will do a few rooms that are back to back or find a better place to tie the electrical and plumbing into. Now whenever you tie old galvanized plumbing up to copper you need a special union so the different metals do not eat each other (yes there are technical terms just not using them here )
SO if it were me, I would clear out the room below, maybe put down an old mattress that I would not use, grab the concrete saw and jackhammer and tear out the floor. I would not worry about the plumbing or stuff in the floor too much (unless there is gas lines) make sure to turn off the water first if there is a possibility there is plumbing in the concrete and same on electrical. Because you mentioned wanting to upgrade the radiators, plumbing and most likely the electrical I would gut the walls to the studs so you have access to the plumbing and electrical, running the new stuff will be easier and it will allow you to see if there are other issues to address. This is why I said first you have to decide how far you WANT to go, if you start opening walls you will find more stuff, which I think is good but it means more time and money. Also if you cut a old galvanized pipe, many times it will let some of the rust in the lines loose wreaking havoc to faucets down the lines ( you will have to remove the screens and run them full open to clear the lines) it also leaves a higher chance of a pinhole leak to start. If you start cutting do not be surprised that you will have to replace most of it in the end. Again I see this as a fine problem, if it is going to spring a leak at any time I may as well start replacing it with newer better plumbing.
Now you are going to say this does not apply to me because I have copper, well I wish that were true. Copper does not corrode but the corner fittings can erode, if they were not properly de-burred when installed the little pieces of copper come off and race down the pipe like sandpaper slowly eating away at the fittings until they leak or burst. Most plumbing systems have a useful life of about 40 years, some more some less, and many are in place much longer with no issue, so dont read this and think your house is going to flood. It is just something to think about when deciding how far to take the project and what to focus on.
I have started using Wirsbo Pex and I like it a lot, I do not have open flames against 100+ year old dry wood, there is almost no chance of the fitting leaking when installed, it can freeze solid and be fine once thawed. Only time will tell if it is viable, but for now that is what I like. You dont need to worry about the new pipes being eaten or destroyed by the old ones and it is made to be easily cut and spliced so when you get to the new section you cut the Wirsbo back, and start again.
Now that I am completely off topic (sorry) yes you can jut go to the hardware store and rent a machine to chip up the tiles off the concrete. It has a blade that vibrates and you just walk the machine forward, works great. You can sledge hammer out the old concrete, either it will have become very hard over the 100 years or it will have become brittle. I like power tools so I would jackhammer it out. I would take out the tub, the surround and replace all of the dry rot as well.
OK, I will get off the soap box now , good luck on your project. I am just amazed to think they have a 4 thick slab on the second floor
most sidewalks are only 3.5 thick.