How can I get to all of these places while studying in London?

From London to Amsterdam, which is one of the shortest travel distances in your list, takes one hour on the plane or three hours by train, but it often takes half a day from door to hostel room.
The other distances are longer and the longer flying times will not add that much, but the distance from the airport to your hostel might add more time.
So if you leave Friday at the end of your lessons, you will arrive at your destination in the evening, sometimes as late as midnight.
On the way back you may need to leave mid afternoon to make it home at all, sometimes earlier.

And while there are cheap flights in Europe, Friday afternoon and evening and Sunday afternoon and evening are mostly more expensive as many people want to go away for a weekend.

I think you might be better off to combine several destinations into two weeks and go for a holiday. Find three locations that are no farther than 3 hours (day travel) or one night (overnight train or bus) from each other and fly out to one and return from the other.
Oslo, Stockholm and Copenhagen can just make it into one trip, but I would not bother with Oslo as the west of the country is so much better. Instead combine the other two with Hamburg or some other northern German city.

Or do Prague and two German cities. Or better still, one German city and a bit of countryside.

Remember that Europe will still be there for many holidays after you end your studies.
If you spend all your weekends and the reading week 'away' you will not have much time to study and that is why you are in London.

They have airplanes in Europe. It's no more than a 2 hour flight from London to Prague, 2 and a half to Stockholm. Book a flight, book a hotel, and go. Have you looked at a map? Checked how close together they are? If you are only going for a weekend, you won't spend much time in either of any two cities. I'd pick one per weekend.