It's hard to tell from that video - a side view would have been more useful.
From what you say it seems the rim is actually round in both planes - it doesn't wobble side to side or up and down as it rotates, so it's not that (but I can't see the rim in the video so I don't know for sure).
If the problem is the tyre it will be either because the tyre is not seated properly, or it's defective/worn out. But that sort of dip is unusual for a worn out tyre - normally they bulge (then pop). If the tyre isn't seated properly you'll normally be able to see it when you look at the side of the tyre. Where the dip is the normal markings on the side of the tyre will dip in too, meaning the bead has been pulled in on the rim rather than being pushed out against the sidewall/braking surface. If you're lucky that will have happened because the tyre wasn't installed properly. This is probably only on one side of the wheel, so check both sides. If it's on both sides probably the tyre is the wrong size for the rim, or you may just be particularly unlucky and have a slightly oversize rim and a slightly undersize tyre.
To fix, mostly deflate the tube, then work round pushing the bead in on the rim as though you were about to remove the tyre. Then pump the tyre back up slowly until the bead pops back against the rim. When this happens stop pumping, and make sure the bead has popped back all the way round. if not, work the tyre with your fingers where it hasn't popped - just squeeze the tyre in with your fingers. The movement should be enough to seat the bead properly.
But if it's a size problem you will need to buy a new tyre. If the size should be right but isn't, try a different brand of tyre. Some manufacturers are known to make "tight" tyres (slightly undersize beads), others do the opposite. Be aware that it is sometimes possible to put a 700c tyre on a 27" wheel (700c = ISO 622, 27" = 630 or so. ISO is roughly diameter, so the difference in circumference is 25mm or about an inch). It's very, very difficult to do, and shouldn't be done at all. If you're unlucky, you have one of the weird size wheels that's almost a modern size. Sheldon Brown discusses wheel sizes here
If the problem is the tube it'll be slightly harder to detect. Tubes are not every strong compared to the tyre, but if you somehow tie a knot in the tube that will affect the shape of the tyre. The most common problems here are a twist, which basically turns the tube into one of those balloon animals; and getting folded back on itself. The fold usually happens when the bike is ridden with the tyre very under-inflated or flat. The whole tyre rotates slighting on the rim as you brake or pedal, but the value stops the tube rotating with it. Until it breaks off, anyway. So you get a stretched part of the tube, and the other side of the value all the slack collects and folds up. When you re-inflate the tyre there's wobble like in your video.
I describe that first because it's easy to check and fix. Deflate the tyre, pop it off, and if there's a big lump of folded up tube next to the valve, there's your problem. Often you can tell just by deflating the tyre than pressing down around the valve.