Soy or almond milk will be comparatively watery, so you'll lose the some of the luscious creaminess of the original drink. Also, since you mentioned trying to avoid heating your non dairy milks, soy milk does taste terrible if heated, but almond milk and coconut milk taste fine when heated. (I'm talking about the the kind of coconut milk that's intended as a milk substitute and comes in cartons in the dairy cooler at the supermarket, not the kind that comes in cans.) But if the only heat source in your drink is the shot of hot espresso, I don't think the heat would be an issue even with soy milk.
I suggest using a liquid non-dairy creamer. It won't be quite as thick as the sweetened condensed milk, but it should be closer than any of the non dairy milks. You have two options - the shelf-stable, unrefrigerated kind (eg, Coffee Mate brand),
and the refrigerated kind that you find in the dairy cooler next to the half and half.
I expect that the refrigerated kind is a higher quality, but I haven't done a side-by-side comparison.
From my quick google search for the example images above, I also learned you can get coconut-based liquid creamers. That sounds delicious to me, but there are many people who hate the taste of coconut so it's not a one-size-fits-all solution.
There is also canned non-dairy evaporated milk, available in the baking section of the grocery store next to the sweetened condensed milk and regular evaporated milk.
Honey and sugar can be tricky to dissolve in cold liquids. If you mix the drink in a shaker, you may be able to get them dissolved with vigorous shaking. But a much more foolproof way to sweeten a drink is with simple syrup. Combine equal parts sugar and water in a saucepan and simmer for several minutes until the sugar is dissolved. Once cool you can store any leftover in a bottle in the fridge.
In my opinion, none of the nondairy substitutes will taste quite as good as the original sweetened condensed milk. So you might want to adjust the proportions so the final drink relies less on the condensed milk, and more on the avocado, for its creaminess. Also, you could add some banana. It'll add a different flavor element, but bananas will definitely give a blended drink that thick, creamy texture you described.