Using a component pouch
A component pouch is not something you pull out and hold in your hand; it's attached to your belt:
A component pouch is a small, watertight leather belt pouch [...]
When you want to cast a spell, you flip it open and grab the component, all one-handed, but the pouch stays put on your belt. If you're wearing the pouch on your belt, it doesn't occupy your hand before or after casting, and it doesn't require an object interaction to access.
If your pouch is inside a pocket or another bag or something, you may need an object interaction to use it, and potentially more because you haven't secured it to your hip the way it's intended to be. A DM might rule that you need both hands, one to hold the pouch and the other to extract a component and do the casting. (I would personally tend to be more flexible about this and rule that just having the component pouch on you is enough to justify casting freely, but I don't like micromanaging the exact position and usage of each of your hands at all times, as long as you aren't telling me you're holding five things all at once. Some DMs are more hard-nosed about the details of what you're doing with your hands.)
So if you have a two-handed or versatile weapon and the War Magic class feature, in one turn, you could:
- Remove one hand from the weapon (free)
- Reach into your component pouch (part of casting)
- Remove the necessary component (part of casting)
- Cast the spell (an action)
- Drop the component you just used, if it wasn't consumed by the casting (free)
- Place your hand back on the weapon (free)
- Make a weapon attack with both hands using War Magic (bonus action)
Using a focus
The situation with the focus is less obviously clear. Readying a focus seems similar to drawing a sword or knife, but that isn't directly mentioned anywhere. Based on the errata'd spellcasting component rules under "Material", it sounds like you need to draw/ready a focus prior to casting:
A character can use a component pouch or a spellcasting focus in place of
the components specified for a spell. [...] A spellcaster must have a hand free to access a spell’s material components - or to hold a spellcasting focus - but it can be the same hand that he or she uses to perform somatic components.
It seems likely that readying a focus counts as an object interaction and precludes a second one -- so you can't put the focus away immediately. The pouch leaves your hand free afterward, but a focus does not, and pulling your focus is probably an object interaction, so if you want to use your hand for something else later in the same round, you should go with a component pouch.
Can you even use a focus?
But a focus may not even be an option in your particular case. The Eldritch Knight does not have a class feature that makes them proficient with any kind of focus, which makes a component pouch your only option (or, y'know, the actual components themselves). You'd have to multiclass into Wizard get focus proficiency for wizard spells, and I'm not sure that would even count; it's somewhat ambiguous whether or not Eldritch Knight spells count as "wizard spells" as required by the wizard's Spellcasting Focus feature. (The spells come from the wizard list, but were not gained through the wizard class.)