Oh my gosh, there are so many variables in this type of task that I don't know where to start in giving advice. First off, the detector circuit in that example will not be useful at all since I believe it operates best at fairly low frequencies and so will trigger from any charge pickup due to mains electrics, MW broadcast, static electricity, and hundreds of other low frequency sources. I suspect it would not operate well above 10 MHz in any case.
Detecting electric field, magnetic field, or electromagnetic radiation at specific frequencies, as the others have pointed it, is relatively easy. Measuring, as opposed to detecting, is very difficult. By measuring, we mean determining the amplitude of the field against some accepted scale or relative to a reference. This implies a certain amount of accuracy, and this is where a hobbyist's attempts will not succeed, as accuracy in measuring fields is a science and an art that takes a lot of effort and experience to get to useful levels. But, if you want to try measurement and aren't too worried about accuracy, it might make a useful and fun project anyway.
The best way to measure RF levels simply and quickly for mobile phones is to get your hands on an evaluation board for the Analog Devices AD8362. You may be able to purchase this directly from Analog Devices using part number EVAL-AD8362EB as noted in this link:
https://www.analog.com/UploadedFiles/Evaluation_Boards/Tools/609195956AD8362EB_0.pdf
(heres the general info on this chip **broken link removed**)
This board is valuable as it is unlikely that you can make the chip work to specification without some RF design tools and experience. If you were to copy the eval board layout exactly and use exactly the same parts they do, then you could build your own. Better off to get one of theirs, then you know it will work.
You would feed this board using specific antennas for each frequency band. Mobile phones will be transmitting somewhere in the range of 824 to 915 MHz if low band, or 1710 to 1980 Mhz if high band. A simple half wave dipole made out of some wire mounted to an SMA coax connector will work but you have to make one for low band and one for high band. In operation, you have no choice but to connect one or the other and measure for that band only as automatic switching between the two is beyond the reach of the beginner hobbyist. Determine the length of the wire (use #18awg solid copper) by calculating the half wavelength in free space and then multiplying by 0.4.
The antenna will be built around an SMA connector intended for pcb mounting. The input to the detector board will be an SMA coax connector. So you will need an SMA-SMA coax cable to connect the two. Keep it as short as practical, say 3 feet or less is fine.
Half wave dipole antennas have some useful selectivity, so such a setup may be more than enough for measuring fields fairly close to the phone. They will be useless when you get too close though. You need to keep the antenna further than about 3 wavelengths from the phone or your measurement antenna will actually disturb the phone's emitted field enough to mess up your measurement.
Such a setup also is completely useless for measuring how much radiation is going into the user's head. In other words, if you put your dipole a few centimeters from the phone, the levels you measure will simply be wrong. This is because you are connecting two radiators, the phone and your dipole, using near field coupling, and creating a sort of transformer that encourages energy to move across the transformer rather than radiate as it normally would. This coupling will be different than the coupling that normally exists between phone and head, so your result will be quite wrong.
Professionals who measure mobile phone levels at or into the human head use a fake head made of a hollow fibreglass mould filled with electrolytic fluid. They put an antenna inside that and move it around to measure field strength. This antenna is an electrically small antenna (not a half wave dipole) that has little effect on the measured field. Its quite complicated really.