First shift up after shifting down only moves as I'm releasing the lever not on both the click in and click out of the lever like the consecutive ones Video of Issue: https://imgur.com/nY8s4Z0

I have just been given my dad's old bike and was looking at fixing it up. However, it has a shifting problem where it shifts down the gears perfectly fine but on any gear, the first upshift after a downshift only shifts halfway.

I have noticed that when it shifts properly when you click the lever in it moves a little and then when you release it moves the rest of the way this however doesn't happen on the first upshift after downshifting.

You can see and hear in the video when I click it in the first time nothing moves and then when I release it moves halfway. Then on the next upshift it moves on both the click in and the click out of the shift lever. Then I move it back down and it shifts down fine on both downshifts but then the first shift up only moves when releasing the shift lever and the second one moves on squeezing the lever and releasing.

I noticed the first click in sounds different to the rest as well almost like it's trying to click it and just slipping.

I would take it to a bike shop but due to the lockdown they won't be open here for at the very least another 4 weeks and id like to be able to ride it in the meantime.

I have tried indexing it at least 10-15 times and all of the consecutive shifts up and down work perfectly its just the first shift up after shifting down doesn't. I watched heaps of tutorials I could find but haven't seen anything about this half shifting problem anywhere.
Not really anything wrong there. You'll want to slowly pedal (with your hand!) while working on the derailleurs. Shifting like this simply creates more and more tension until it's unable to move the chain anymore sideways. Chains can only move so far sideways.

Also make sure the cable is free to move easily within the housing. Dropping a bit of (chain) lube at the ends and pulling it into the housing can already help a lot.