Supposing Jesus will judge the living and the dead, is this judgment an omniscient one?

For the Father judgeth no man, but hath committed all judgment unto the Son: [John 5:22 KJV]

If God the Father has committed all judgment unto the Son, then, yes, this is an omniscient judgment. Why would you suppose it was otherwise ?

As to what God, the Father, and God, the Son, knows :

I know that thou canst do every thing, and that no thought can be withholden from thee. [Job 42:2 KJV].

Every one of our thoughts is transparent to Him who is Spirit.

God is a Spirit: [John 4:24 KJV]

Short answer: Yes. The judgement of Jesus is an omniscient one. There is no difference between being judged by Almighty God and being judged by Jesus Christ.

Longer answer:

And the kings of the earth, and the great men, and the rich men, and the chief captains, and the mighty men, and every bondman, and every free man, hid themselves in the dens and in the rocks of the mountains; And said to the mountains and rocks, Fall on us, and hide us from the face of him that sitteth on the throne, and from the wrath of the Lamb: For the great day of his wrath is come; and who shall be able to stand? - Revelation 6:15-17 KJV

Is there anything more incongruous to the human intellect than the wrath of the Lamb? And yet, understanding and believing who this Lamb is and what He has done is the beginning of wisdom.

In the Epistle to the Hebrews we are told to remember the Israelite's wanderings in the wilderness; to remember that most of them perished without ever having entered into the "rest" that they were promised and, because of this remembrance, to fear:

We may fear, then, lest a promise being left of entering into His rest, any one of you may seem to have come short, for we also are having good news proclaimed, even as they, but the word heard did not profit them, not being mixed with faith in those who heard, for we do enter into the rest—we who did believe, as He said, 'So I sware in My anger, If they shall enter into My rest—;' and yet the works were done from the foundation of the world, for He spake in a certain place concerning the seventh day thus: 'And God did rest in the seventh day from all His works;' and in this place again, 'If they shall enter into My rest—;' since then, it remaineth for certain to enter into it, and those who did first hear good news entered not in because of unbelief— Hebrews 4:1-6 YLT

We are also told to realize that the 'rest' that Joshua led them into (the promised land) was not the ultimate 'rest' but merely a type of that rest which awaits us. The promised land was already flowing with milk and honey for God's people to partake of without any effort on their part. Even though there were enemies to conquer God had declared that he would hand them over. We see, for example, the Israelites victorious through faith (trusting in God's finished work) at Jericho and failing through unbelief at Ai (trusting in their own ability is mistrusting God):

again He doth limit a certain day, 'To-day,' (in David saying, after so long a time,) as it hath been said, 'To-day, if His voice ye may hear, ye may not harden your hearts,' for if Joshua had given them rest, He would not concerning another day have spoken after these things; there doth remain, then, a sabbatic rest to the people of God, for he who did enter into his rest, he also rested from his works, as God from His own. - Hebrews 4:7-10 YLT

Knowing this we may acknowledge another, ultimate and true Sabbath Rest which is still available for the people of God which is to be entered by faith:

May we be diligent, then, to enter into that rest, that no one in the same example of the unbelief may fall, for the reckoning of God is living, and working, and sharp above every two-edged sword, and piercing unto the dividing asunder both of soul and spirit, of joints also and marrow, and a discerner of thoughts and intents of the heart; and there is not a created thing not manifest before Him, but all things are naked and open to His eyes—with whom is our reckoning. - Hebrews 4:11-13

Here we see that the "reckoning of God" discerns the thoughts and intentions of the heart. This "reckoning" is, properly, Logos ... the Word. It is that same Word which was with God, was God, and was made flesh:

In the beginning was the Word, and the Word was with God, and the Word was God; - John 1:1

Jesus Christ is the "divine mind", the "reckoning of God", made flesh to redeem flesh, come as a man to bear human sin and provide forgiveness to man, born under the law to redeem those under the law. He has the ability to judge and execute justice because He is the very reckoning of God. He has the authority to do so because He is the son of man. There is no difference between being judged by Almighty God and being judged by Jesus Christ.

Those who have given up pursuing righteousness via their own works and rely utterly on the righteousness of God which is by faith of Jesus Christ have entered into that rest and have nothing to fear in the judgement. They have passed from death unto life and will never be condemned

Those who have not yet applied to the crucified Christ to take their sin/death and be given life or those who have only done so waiveringly (still trusting in religious activity and a righteousness of works) have everything to fear of this judgement because it is wrought at the level of thoughts and heart intentions, not mere deeds. They are condemned already. And yet while they still live there remains this one great hope:

Wherefore (as the Holy Ghost saith, To day if ye will hear his voice, Harden not your hearts, as in the provocation, in the day of temptation in the wilderness: When your fathers tempted me, proved me, and saw my works forty years. Wherefore I was grieved with that generation, and said, They do alway err in their heart; and they have not known my ways. So I sware in my wrath, They shall not enter into my rest.) Take heed, brethren, lest there be in any of you an evil heart of unbelief, in departing from the living God. But exhort one another daily, while it is called To day; lest any of you be hardened through the deceitfulness of sin. For we are made partakers of Christ, if we hold the beginning of our confidence stedfast unto the end; While it is said, To day if ye will hear his voice, harden not your hearts, as in the provocation. - Hebrews 3:7-15 KJV