Most sewage ejection pumps have two cords. One is the power for the motor. The other goes to the float switch. The power cord is a regular plug. The float switch cord has a plug on the end with female on one side and male on the other. The power's plug is inserted into the female side of the float switch plug. The male side of the float plug is then inserted into an electrical outlet. When the water level trips the float, the float's plug "opens" to give power to the motor's cord. If you want to test the pump manually, find the outlet the cords are plugged into, unplug the motor's plug from the float's and insert the motor's plug directly into the outlet, bypassing the float. When you're satisfied, insert the motor's plug back into the float plug and insert the float plug back into the electrical outlet.
My guess, and it's only a guess, is that that pipe is where the downspout used to attach to the sewer system. Until very recently, downspouts used to connect directly to the sewage system. Many municipalities have passed laws mandating the removal of these connections because the treatment plant would often overflow during a storm, leading to the dumping of untreated sewage into natural waterways.
So what's happened, is there is a block downstream of the connection between this storm sewer and the household sewer. Typically, if the blockage occurs on the road side of the water supply cut off --
-- then the city is responsible. If it's inside, then the homeowner is responsible.
In my city, they'll run a camera down the pipe, and charge you $300 for the scoping if the blockage is on your side. I'm betting in the short term, you'd be better of to get a city licensed plumbing and drainage contractor to snake it out and absorb the cost. Not a DIY job, because the city will make you pay for anything that goes wrong.
The next issue is how to stop it from happening again. You'll probably have to dig down to the connection, remove the downspout and cap it off. Again, since you're playing with municipal systems, this is not a DIY job.
You might get lucky with calling the city, but then again, they might inspect certain systems and order you to fix them right away, at a cost you likely can't afford if you just bought the house.
Best Answer
The answer is NO:
electrical wire has to be used in an approved manner.
UF is approved for direct burial and use in an approved wireway or other listed methods.
a sewer pipe is not an approved method in any of the articles associated with NEC 340.10: used permitted for underground feeder type UF.
the answer is NO it is not permitted.