One who does not recite Bismillah before wudhu, he will not receive the full reward of his wudhu, though the wudhu will be valid.
عن سعيد بن زيد رضي الله عنه قال: قال رسول الله صلى الله عليه وسلم: لا وضوء لمن لم يذكر اسم الله عليه. (ترمذي رقم 402)
Hadhrat Sa’eed bin Zaid (Radhiyallahu Anhu) reports that Rasulullah (Sallallahu Alaihi Wasallam) said: “A perfect wudhu will not take place for the one who does not take the name of Allah Ta’ala (at the time of commencing the wudhu).”
عن ابن عمر رضي الله عنهما قال قال رسول الله صلى الله عليه وسلم: من توضأ وذكر اسم الله على وضوئه كان طهورا لجسده ، ومن توضأ ولم يذكر اسم الله على وضوئه كان طهورا لأعضائه (السنن الكبرى للبيهقي رقم 202)
Hadhrat ibn Umar (Radhiyallahu Anhuma) report that Rasulullah (Sallallahu Alaihi Wasallam) said: “The one who makes wudhu and takes the name of Allah Ta’ala (at the beginning of the wudhu), the wudhu will be a means of purification for his entire body (i.e. for the minor sins committed by the entire body), and the one who makes wudhu and does not take the name of Allah Ta’ala, the wudhu will only be a means of purification for the limbs of wudhu (i.e. for the minor sins committed by the limbs of wudhu ).”