Hey,
you sat down without discussing this very frequent bidding? How careless.
I would take 4
♣ as a shortage. It SHOULDn´t be and with all my regular parts it wouldn´t be, but without discussion it is.
5
♦ shows a second round control and nothing in Hearts, but obviously something extra, else he could have bid 4 Spade.
5 NT is still a grand slam try. After you establish Spades as trump, a "pick a suit" is just wastage of bidding space. But I think, it must be a very special Slam Try, called josephine. You had no good bid to ask for Spade Honours after the bidding surprisingly went on to 5 Diamond, so this should ask for Trumps.
6 Clubs is the smallest possible answer, so I guess, he denied any spade Honour. This is impossible, so I now I must guess the final contract and guess 6 Spade.
I think the bidding was fine till 4 Club. With all side aces, with a source of tricks and with a not well established partnership, you should just bid 4 NT, RCKB and placing the contract depending on pds answer. 4 Diamond led to nowhere.
1♠ p 2♣ p
2♠ p 3♠ p
4♣ p 4♦ p
5♦ p 5NT p
6♣ all pass