How do I make a cucumber homemade ice cream using all-purpose cream as whipped cream?

Yes, you need a crème anglaise base because those egg yolks are going to help "elasticize" your ice cream mouth-feel.

It's possible I am the only chef on earth who likes cucumber seeds, but I think they would help your cause. I don't think the water in your cucumbers is the issue, and I don't suggest getting rid of it.

This is what I would do if I wanted cucumber ice cream:
Make anglaise with 1Q cream + 1Q 1/2 & 1/2 + 16 egg yolks (from large eggs) + pinch sea salt (not iodized) + 1 Cup sugar.
Peel cucumbers. Make sure you have at least 16 ounces (about 2 cups). Puree as fine as you can (a vitamix is best.)
When anglaise is cool, whisk in cucumber puree to taste. Do not add vanilla extract or flavorings of any kind, until the flavor is where you want it on the base.

Make sure you're watching the ice cream when you churn it. If you over-churn it, it will get grainy and hard. Best to under-churn it a bit.

If it's still too hard, melt it down and whisk in some Lyle's Golden Syrup or Glucose, and maybe even a little vodka or white wine.

Sugar and alcohol lower the freezing point of water.

I hope this helps. Best of cucumber luck!

What is all purpose cream, anyway?

I make it using a custard base. Here's the recipe: https://food52.com/recipes/4936-cucumber-mint-ice-cream. If you prefer not to use eggs, try using Jeni's ice cream base and adding the pureed cucumbers. https://food52.com/recipes/14257-jeni-s-splendid-lemon-cream-ice-cream

I agree with Nancy. There is possibly something wrong with the proportions you have. It is important to note that the cucumber is 90% water so that may be why you are ending up with more ice than cream. One thing you could try is salting the cucumber and allowing it to exude some of the water which it will invariably do and then rinsing it. And maybe proceed from there. I would also recommend that once blended you wring out any moisture from it before moving on to the next step. I also noticed that you are not using any eggs which would add some full bodied creamy texture if you are comfortable making a custard to which you would then add some cream infused with the cucumber. Best of luck...ice cream can be so darn tricky!

No expert on ice cream, but I've made it at home several times.
Can't diagnose or remedy your specific recipe, but guess there are problems in the amount of fat and proportions.
Suggest you make some tested recipes for cucumber ice cream and see what results you get.
There are a bunch in this search. Better start with those that come from reliable sites with testing or ratings, rather than individual bloggers - whose results may be good or uneven.
https://www.google.ca/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=cucumber%20ice%20cream