I'm a bit concerned that you are riding on a bike in that condition.
I have no experience of this problem. But I have been thinking about it. This is just a suggestion.
The two parts that are stuck together are the collar and the bracket. The collar is normally easily removed once the bracket has been taken off. Those two pieces have got to be separated before you stand any chance of getting the bracket out or the wheel out.
Some amount of force is likely to be necessary, but it is best to try to keep this to a minimum. The problem is making sure that you don't damage anything else while you are doing this. In the video, tapping the wood with the hammer is not effective because the collar can move and the wheel can move. Both of those need to be held firm.
Suggestion:
Here is a diagram I put together this morning so that you know which bits I am talking about.
1)
Remove the brake pads. If the wheel, bracket or caliper moves and the caliper is held against the disc by the pads, you will bend the slider pins, or warp the brake disc.
2) Apply some penetrating fluid all the way around between the collar and the bracket (
red arrow). PlusGas is a common one in the UK. It is VERY 'wet' and gets everywhere. Make sure it doesn't get into the wheel bearings. Stuff a rag through the axle hole to soak up any that might get down the inside. Keep the fluid away from the wheel hub. Leave it for a while to do its work.
3) Insert the axle so that the (
yellow) collar is supported by it. Make sure that the axle does not enter the hole in the brake caliper. You will need to measure this to be sure. Measure the width of the swing arm, and add on the width of the bracket. Measure how deep end of the axle is in the hole from where the nut is normally located. The depth of the hole to the end of the axle needs to be slightly more than that total. The diagram shows the axle in this position.
Note that the collar (yellow) is supported by the axle, the bracket is not.
4) Support the rear wheel. The wheel is off the ground - use chocks, or drive a piece of wood under the tread to stop the wheel from moving. This takes the strain off the axle which is supporting the entire weight of the wheel, just on its right hand end.
Now, the only component that can move is the caliper bracket. The wheel should stay firm. The distance collar should to stay firm.
There are now a few ways that you can try to shift the bracket:
One way is to knock it free. You may need to remove the caliper bracket stopper bolt - although the hole is oval, so it may not be necessary. The bracket only need to move a fraction for it to break free. You might be able to tell if there is space for it to move by looking through the wheel from the right hand side. If the bolt is in the centre of the oval, then leave the bolt in. It will prevent the bracket from moving too far and putting strain on the brake hoses.
The second way would be to arrange a jack (a bottle jack perhaps) underneath the caliper bracket (
blue arrow). It would probably need something between the jack and the bracket itself. If you can arrange this, then wind up the jack until the bracket is supported and then wind it up a fraction more - just so that the weight of the rear end of the bike is being taken on the bracket. Not much - you are not trying to lift the wheel - just taking the weight. Leave it for a while, let the weight do its work. Remember that when the bracket breaks free, the bracket will stay where it is and everything around it will drop a little bit. For this, the stopper bolt can certainly remain in place.
A third (that I have just thought of) is to use the small bevel that is on the collar to try to drive a wedge between the collar and the bracket. I'm not sure if the bevel is at a suitable angle for this, but it may work.
These are only suggestions. Think about it before you do it. As I said, I have never come across this problem before.
I hope it helps.