To provide some more data points, here's my experience:

Registered at Minato City Hall the day after I arrived; got a temporary certificate (which has the date to collect, which was indeed three weeks distant) and also several copies of the certificate of registered matters. It's 300Y per copy but it's worth having a few.

From there, I went to the Tokyo Immigration Bureau to get my multiple entry stamp and register for the automated gates at NRT. 6000Y for the multiple entry stamp; buy it from the convenience store downstairs before queuing up.

That afternoon, I went to Shinsei Bank in Roppongi Hills (6F of the Mori Tower) who told me I couldn't open a bank account until I had a phone number. Since I didn't actually have a permanent extension at work yet, I went to get a cellphone.

I went to the SoftBank store at Roppongi Crossing and they only needed my passport and a copy of the certificate of registered matters to send me on my way with a phone. They had at least two good English speaking staff and I got the impression they're used to expat types coming straight from the ward office without their cards yet.

Back to Shinsei who opened my account straight away - but for security reasons they had to send my cash card through the post (they don't do this if you have the actual alien registration card). It arrived three days later and the internet banking card the day after.