Curve 25519 (X25519, Ed25519) Convert coordinates between Montgomery curve and twisted Edwards curve
The formulas actually work.
You just have to keep in mind to make computation in the field of intergers modulo $2^{255}-19$ and that there are actually two square roots, you need to use the right one if you want to have the expected result.

You can test the following SAGE code

gf=GF(2^255-19)
X_P = gf(15112221349535400772501151409588531511454012693041857206046113283949847762202)
Y_P = gf(46316835694926478169428394003475163141307993866256225615783033603165251855960)
u = (1+Y_P)/(1-Y_P) 
v = gf(-486664).sqrt()*u/X_P,-(gf(-486664).sqrt())*u/X_P
print u
print v


on the Sage Cell server.