The question says it all. Besides Russell and Whitehead, who are the philosophers and mathematicians who demonstrated successfully that 1+1=2? Extra bonus (i.e., a warm feeling inside) awarded for identifying the book or essay where the proof appears.
In a modern axiomatic approach, it follows from the definition of 2 as the successor of 1 :
2=s(1) (also written : 2=1').
See Giuseppe Peano, Arithmetices principia: nova methodo exposita (1889), page 1.
The basic machinery was already in place in Richard Dedekind, Was sind und was sollen die Zahlen ? (1893), para.XI : Addition of numbers :
sum is completely determined by the conditions
m + 1 =m',
m + n' = (m + n)'.
Applying the first condition to m=1 we get :
Peano's approach is presumably derived from Leibniz's well-known proof of 2+2=4.
See New Essays on Human Understanding (1704), Book IV, vii,10 [English transl, page 414] :
Definitions. (1) Two is one and one.
(2) Three is two and one.
(3) Four is three and one.
Leibniz introduces only one axiom for equality :
Axiom. If equals be substituted for equals, the equality remains,
and thus the proof is incomplete, by modern standard, because it relies on the implicitly assumed associativity of sum.
Another source of Peano's work has been Hermann Grassmann.
Grassmann's Lehrbuch der Mathematik (1861) contains a reasonably complete axiom system for arithmetic.
See Hans-Joachim Petsche, Hermann Grassmann : Biography, Birkhauser (2009), page 198-on.