Fallout – Where is this in Fallout 4

After watching the video I'm certain it was just your follower walking past the cell which caused it to open. Also the creepy background music seemed to already be playing, and wasn't triggered by the door opening.

The doctor in Diamond City was the first one I ran into in my first couple playthroughs. There are a few doctors you could reasonably get to beforehand without going way out of your way, but they're neither a lot sooner nor much easier to get to, so you aren't missing out by just going ahead and continuing to Diamond City instead.

Bethany

There's a doctor in a shack surrounded by radiation west of ArcJet Systems. The doctor is a ghoul, so the radiation doesn't bother her. If you already have enough radiation resistance, perhaps from the power armor in Concord, then this doctor might be worth visiting. Unfortunately, there's no fast travel waypoint.

Patricia Montgomery

There's a doctor in Covenant. Covenant is pretty easy to reach if you go east a bit. You'll have to go through the SAFE test at the door, and she may turn hostile depending on your choices in Human Error.

Rachel

There's a doctor you can see in Vault 81. I often go by Vault 81 on my way to Diamond City, but it's not very far away and you may not even need to fight anything between the two locations, so you're not really gaining much by going here first.

Furthermore, you have to either pass a charisma check or donate three fusion cores to gain entrance. The check isn't too hard, especially if you don't mind loading a few times. You should be able to get past it with some Charisma gear and/or some temporary buffs.

Doc Weathers

There is a traveling doctor. I usually run into Weathers at Bunker Hill, in the stall next to Deb. This stall rotates between the various traveling merchants. Sometimes Trashcan Carla is there. Weathers can also be found traveling around the northern part of the map. The wiki article says he starts at County Crossing, near the National Guard Training Yard.

If you spread east before going down to Diamond City, you'll probably run into Weathers at some point. Bunker Hill is a fairly easy to reach early hub. You could just as easily get to Diamond City though, if you decided to go that direction instead.

Make Your Own

You'll probably get to Diamond City before Level 14, but maybe not, depending on play style. At level 14, you can get Local Leader 2. This, combined with Medic 1, allows you to build a pharmacy in your settlement. You can assign a settler here, and get all the services provided by a normal doctor.