I used to be local - I'm less local now. My family still live up there though and we go up quite frequently.
I would assume that the fountains will all be on. My dd and I went to the gardens in the Autumn and it was freezing, but everyone was still messing around in the fountains! There's this one big one where the water shoots up from the floor and you can stand inside the ring where the water shoots up while the water is filling up, then it shoots up in front of you. In theory, if you stand still, you shouldn't get too wet, but all the kids always run in and out and get soaking!
If you are going to the castle as well, then it is (slightly) cheaper to get day tickets for both the gardens and the castle at the same time - for adults at least. You can walk from one to the other. Also, the parking near the castle is terrible - actually, parking in most of Alnwick is terrible! You should be ok parking at the gardens though, they have their own parking there.
Alnwick castle is quite child friendly and kids can usually dress up as witches and wizards and take part in some activities. I doubt you will spend an entire day at the castle though.
Depending on the age of your dc, there is a soft play in Alnwick (if you find yourself bored at any point. You might not, but having lived there for the majority of my life, I get bored easily!) Link:
The Lions Den
I don't really know Seahouses that well. I did work up there while taking my A levels (in a fish and chip shop ) there is a small crazy golf course and loads of annoying amusement arcades!
Holy Island is one of those places we were forced to go for a school trip as children. I suppose it was fairly boring. We just walked around and looked at the priory and the castle and bought some cheap souvenirs.
That's the problem with Northumberland generally. It's very much a walking round and looking at stuff sort of place. Although very pretty, it can be quite boring for kids. At least there are loads of good beaches. You can't go wrong with a beach!
I hope you enjoy yourselves :-)