Should I skip past a few months or play it from the escape?

I'd suggest playing through the escape, you can do it real quick if you want, and then if you want you can do a time skip. You know the classic "Three Weeks Later" and put them pretty much wherever you want in DC.

Or you can make the trip from here to there part of the story. Whatever floats your boat.

Ok, I will try to write the next part after School and then I have the full weekend

A piece of advice, with the weekend coming up, even if you finish a bunch of stories in close-time, I would put some time in-between each chapter posted.

Chapter V - Cold

We walked for about an hour, walking was becoming a great effort for Tim even with our help. We eventually found a place beside a lake under a collapsed bridge, Tim was bleeding bad. There was nothing we could do except stay with him to keep him fed and hydrated, none of us had any medical experience. We sat next to Tim trying to keep him alive while he choked his own blood. He tried to talk a few times but was unable to gather the strength. It was all my fault, all those deaths in the Vault were my doing, that's what I get for trying to help people I suppose. I looked to Tim, he was crying.

I felt a tear stream down my eye, I had not cried since I woke up in Vault 87. I never knew who captured me and took me to Vault 87, neither does Tim, he deserves to know that it was me. I'm not even sure if he could hear me but I gathered the strength to tell him.

"Tim... you don't know who took you to Vault 87 and made you what you are do you"?

Tim didn't respond.

"Well, it was me, I am the reason you are a Mutant, I am the reason you are where you are now, I captured you and took you to Vault 87".

Tim didn't respond yet again, but it wasn't because he couldn't hear me.

Tim was dead.

The world had never felt so cold.

A short and sweet chapter, but I don't think you should have killed off Tim. I would've given him quite a bit more character development before dropping him from the story.

In most cases this would have been a good edition, but we aren't introduced to very many characters, so I'd recommend holding onto them until we really get to know them, and then drop them. Makes it more moving or exciting for the reader.

Thinking of them travelling to the Mojave to meet other mutants

(That's a hell of a walk, so you'd have to make a hell of a time-jump)

I'm sure I can make up something good like them hijacking a vertibird or something less rediculous