"тем временем" and "кстати"

The question refers mostly to English speakers rather than local Russians.
What is the difference between "by the way" and "meanwhile".
The simple translation in my head is that both mean: at the same time.
In both occasions the two phrases are used when we want to attach to an initial piece of information a second one that might be related or not to the first.

Example 1: "She was trying to finish the exams on time. Meanwhile, John was trying to find where was his pencil".
Here if I use by the way it doesn't sound correct at all.

Example 2: "Yesterday I cooked beef with potatoes. By the way! I forgot to tell you that George is getting married to Julia"
Here if I use meanwhile, I don't know if it sounds correct.

Can someone provide a "road map" to the explanation of there difference.

I guess "meanwhile" refers to time (only?), like you said "at the same time". But "by the way" refers to some events or situations you have to describe right now, I think because it contains the word "way". That's how I understand it.

By the way, there is the same issue in Russian.

By the way = кстати, между прочим
Meanwhile = тем временем, в то же время, между тем