General Advice: Use a system with simple rules.
The more rules the players have available to them, the more they tend to rely on them. When all you have is a hammer, everything looks like a nail. The less game mechanics the players have to use as tools to "solve" an encounter, the more incentived they are to come up with creative and more narrative ideas.
From what I've heard about D&D 5th Edition, it appears to be a game well suited for this purpose.
Rulings, not Rules
A central part of oldschool gaming is that the rules of the game are tools to help the GM determining whether an action by the PCs is successful or a failure. They are not meant to determine what happens. The consequence of this is that you don't need a preexisting rule to try something. It is up to the GM to decide whether an action be "successful", "possible (roll dice)", or "impossible". When an action sounds like it could work but it could also fail, the GM has to decide what kind of die roll the player should make.
In early editions of D&D these were usually just attack rolls, ability checks, and saving throws. If the action is intended to cause damage, you pick a number of d6 that seems to be roughly appropriate. The more recent edition there's also skill checks, which are just a more customizable form of ability checks. When a player wants to do something and points invested in a skill sound like they might help with pulling it off successfully, make a skill check.
To run the game in a more oldschool way, treat the descriptions of the skills as examples of how the skills can be used. They are not a complete listing of all the ways in which the skills could be used. In actual play it's often a good idea to just eyeball target numbers and describe the result of the action as it seems intuitive to you. Don't get out the book every five minute and spend 30 seconds looking for the exact skill description. That would just slow things down way too much and gets everyone back into the habit of trying to solve problems through calculations instead of using creative narrative solutions.
Obviously it's important that the players know you're doing this and that they are okay with the books not always being strictly followed for the sake of having the game run smoother and more flexible. You're not cheating when you're doing this, it's a valid way of playing the game. But the players need to be aware that this is how this campaign is being run.
When in doubt, always say Yes.
Combat as War is not really about having things be realistic, but about things being awesome. It's about the PCs being badass. You also want to encourage the players to come up with awesome and badass ideas to defeat their opponents. So unless you feel that a proposed action really should be completely impossible, let them try with a die roll. And if you think the action really shouldn't be hard to pull off, just say that it succeeds without rolling any dice. If it should work and a bad die roll would be a disappointment for everyone at the table, just don't roll dice.
Also, every time you say no to something, the game stalls for a moment. The players had an idea but you said no. Now they have to think again to come up with another idea. That breaks the pace of the game and might interrupt a heated action scene. So unless you think you have to say no, say yes. That also goes for any time when the players ask questions about the environment. If a player asks "Is the bridge made of wood", he probably has a great idea of what he could do if the bridge is wood. Unless you have a good reason why the bridge should be made of stone, just say yes. It should be fun.
Always make sure you know what the players are planning
A critical part of making a ruling is that you and the player are on the same page about what the situation is like and what's about to happen. If a player says he jumps over the wall but there's a 50 meter drop on sharp rocks, it's almost certain that the player is imagining the situation differently than the GM. The player's character would know because he can see the environment, but the player himself only makes assumptions based on what the GM told him. So before you make a ruling what happens, always make sure that the players are imagining the same situation as you do.
Know the Goals and understand Dramatic Questions
In a Combat as War scenario, the goal of all involved sides is never to kill all opponents at any cost. Fighting and killing is always a means to accomplish an end, not the end itself. A Combat as War battle is not about killing all enemies, but about getting the enemy into a position where they can no longer achieve their goal and won't have any choice but to retreat or surrender. This goes for both the players, as for their opponents. And sometimes it might even be enough to keep the enemy at bay for a while, accomplish the goal, and then leave the battlefield. If your goal is already accomplished, by keep risking your life?
This leads into the Dramatic Question of every encounter. What is this encounter about? What question is this fight going to give an answer for? There is a really great and extensive article by the Angry GM about this whole topic, which I very much recommend anyone to read: Four Things You've Never Heard Of That Make Encounters Not Suck