Their auctions:

p
p p 2D p
2H p 2N p
4C p 4D p
4H p 4S p
6D

p
p p 2C p
2D p 3C p
3D p 3N p
4C p 4D p
4H p 4N p
6N

First, partner will show good diamonds, meaning KQxxx or better. That's key. If you don't play that way, it's a lot harder.

If partner has an outside king in addition to good diamonds, 7NT is there. If not, we need to play in diamonds, which will work if I can ruff a club; I'll guess that I can, because I need to go above the safe level (3NT) to find out). And if I can get to partner's hand to draw the diamonds, then I'll have twelve tricks exactly if partner has five diamonds, thirteen if he has six. Because we may be off the diamond jack, though, in this case I'll bid six diamonds.

The alternative is if partner has no outside king but has club support, in which case there will usually be a third-round club spot entry to the KQ of diamonds. In that case we'll make six.

2C – 3D
4C – 4D
4NT – (whatever response — doesn't matter as long as it's ace-asking)
5NT – (whatever indicates the king of hearts)
7NT

If 4NT won't be some form of ace-asking on that auction, then as opener I'll jump to it immediately, then ask for kings. Upon hearing about the diamond king I'll bid 7NT.