Gosh, this is really dependant on the house. My family owns a farm, and we have a well...and that answer would depend on how far DOWN the interdimensional rift goes, because the well is pretty close to the house, so it would be included, it's just a matter of how far down it goes.
A place in the country, or in a quirky neighborhood is more likely to have an alt water source, such as:
- Rain Barrels off the gutters
- onsite cistern (either in the ground in the case of residential or on the roof in the case of apartment buildings)
- an actual well. Sometimes happens in a residential area that everyone else will be on city water, but your old house has a well and pump. The problem is that the pump needs electric to work most of the time (though there are some that have a crank pump for emergencies--these are mostly installed by paranoid folk or people who simply want to live off the grid or electric goes out often, or because the old pump wasn't hooked up to the new system, and this is a remnant of the old system.) Without electric, the pressurized tank will supply water for a little while. These tanks range from as little as 2 gallons (mostly for places that are using it for small watering needs, such as periodic watering of plants, while the rest is on city water) to as much as 114 gallons. The standard size tends to be about 20-40 gallons, but 80 gallons can be common. Varies widely!
I looked at the original question and frankly, if the house is just CUT away, including the plumbing, then there will be issues in OUR dimension.
See, mainlines can run through front yards, and then have lines off that, which means that in reality, where the house has disappeared from, there's bound to be problems. Even if things sync up exactly, a pipe can be several feet to the left or right if you cut out the middle where the house is, so it might not match up.
In other words, there's a plumbing problem everywhere....and outside of reality there's not enough pressure in the lines once they are effectively cut, to give you any water. Maybe for a second, but that's it. In the U.S. there's a water heater, and those sometimes have water stored in--depends on the size--they can be 80 gallons, they can be 25. About 50 is the average. Tapping it could be difficult depending on the system.
UNLESS--introducing interdimensional plumbing AND electric! You can't get wifi, (or maybe you CAN!) but you do get intermittent plumbing and electric. Anything with a hard line in or out works, kind of.
Because you're bending space-time, the house is really still there, it's just in a pocket dimension. Maybe nothing organic can get through, but objects and energy CAN. Which opens up a whole world of possibilities...