TL;DR Light Yagami.
I find it ridiculous to assume Light would lose simply because his plan did not succeed. Firstly, he turned into a villain, while Lelouch was an anti-hero [and we all know the disparity in success rates between such archetypes in fictional works]. Secondly, I'm sure I don't need to prove how unrealistic the reaction of most of the world was in Light's world or the lack of L/N-level supernatural intelligence/thinking in our world. Thirdly, mind I remind you that Lelouch got lucky and was rescued by other people several times [The Black Knights, C.C., Rolo]. Fourthly, by such reasoning we may as well just base the result of every battle between intelligence-based characters on whether their plans succeeded or not. Etc.
I'm assuming they are both in-character.
Lelouch is not as ruthless or heartless as Light. He would not put civilians in harm's way from the get-go on the off-chance that Light is one of the casualties or have the desire to repeat the massacre committed by Euphemia li Britannia.
Lelouch has a slight advantage due to the circumstances and practically of Geass in the short-term, however the Death Note's range makes up for it in the long-term.
Lelouch would treat the mission as a judgment for his less than moral deeds, while Light would think of it as a second chance or the resulting judgment/afterlife for Death Note-users as described by Ryuk.
Our Tokyo is more similar to Light's Tokyo than Lelouch's Tokyo Settlement, making the former more at home and less inclined to assume that he has been transported to another world.
The first piece of information they will consider; the opponent has supernatural powers. Lelouch would at the least assume that he was facing another Geass-user, however Geass-users are not immune to each other's powers and Geass can manifest in many forms, so he wouldn't be as worried as Light, who would at the least assume that he was facing another Death Note-user, making it certain that he would reach a near state of bloodlust to be the first to kill the other at the quickest rate possible.
Lelouch would not use a command such as "Kill yourself, if you have supernatural powers!/if you can't leave Tokyo!" from the get-go. For these orders to even have a slight chance of bringing results, he would need to go around all of Tokyo shouting at crowds of people. Firstly, not practical [9.2 million people]. Secondly, Lelouch is not that reckless. Thirdly, that would make it certain that Light would be the first to find him not the other way around. Fourthly, the first command might not even work on Light, as he personally doesn't have supernatural powers, but that's irrelevant. The second command is not very specific and could cause the death of innocents.
Their first priority would be to secure a temporary or permanent residence and informatio. Lelouch can easily achieve this through his Geass. Light would rely on his charisma/manipulation/looks [Misa Amane nr.2]. It is possible for him to manipulate the actions of individuals without instantly killing them by writing a specific long disease as the cause of death, so no he wouldn't need a job to sustain himself or be at a disadvantage at getting himself to the top.
After securing a base of operations, time to search the internet [I'm serious and assuming that Lelouch would be familiar with our technology]. Each of them would search for key figures/terms from their own world, such as Light searching for L [well-known figure in his world], specific organizations etc, while Lelouch searches for Britannia, the current Emperor of Britannia etc. Now they are both positive that they are in another world.
The second and key piece of information they will consider; the opponent cannot leave Tokyo. Time to reach the top.
Lelouch would not create a mindless horde of loyalist civilians. I do not believe I have to elaborate on how reckless this would be for his own well-being [No way the media|officials in Japan would be oblivious to such a threat/happening in their own backyard for too long. It would be only a matter of time before the media starts filming and broadcasting his actions and he is gunned down or identified by Light through the internet and killed(search "absolute obedience superpower")] or the impracticability of the task [Direct eye contact with the entire population (9.2 million) of Tokyo].
Neither Light or Lelouch would consider a direct confrontation between each-other due to the fear of facing an opponent with supernatural powers and the fact that they are used to manipulating others to do the dirty work for them.
Each would search on the internet for the highest-ranking officials, the difference being; Light would search for the highest-ranking officials of Japan and possibly the world, while Lelouch would search for those inside of Tokyo. Compare now the firepower they have to work with and who would be faster.
Recall the arrest of Misa Amane, a Death Note-user. The police avoided eye contact, were wearing headgear that hid their faces and made it impossible for her Shinigami eyes to identify them. They also blindfolded her during the arrest. Each of these precautions are effective against a Geass-user like Lelouch [To add onto the fact that such methods are truly effective, recall the headgear that the soldiers were wearing in the episode "Emperor Lelouch"] and Light has at the least assumed that his opponent could be a Death Note-user.
[I'm aware that the Death Note will not be effective if the death of the first victim leads directly to the death of others by that first victim, but this can easily be circumvented.]
While Lelouch is physically walking towards the residence of the people in power [which in themselves would prove time-consuming/difficult to find and enter], Light starts, moments after searching the internet, writing the names of the highest-ranking official of Japan [with the cause of death being a long-term specific disease] to order them to leave, evacuate Tokyo and send in the military armed with the above equipment with the orders to obey and protect Light Yagami [To specify, due to the rules of the Death Note; the military will be ordered to obey and protect a person that matches Light Yagami's physical description, current location and knows a secret password, so we can avoid Light writting his own name in the Death Note]. If we are going for an overkill, he can also write the name of all the world leaders and order them to send their armies to assist him.
Light is now surrounded by bodyguards and orders them himself to find the other remaining person inside of Tokyo and kill him.
