Wondering about a ruling if I am using the spell Speak with Animals rather than trying to communicate my intentions with the animal via Animal Handling skill for checks for communicating after casting would I use Persuasion and Deception? What are your thoughts is it dependant on e.g. Int level?
My ruling as a DM: let the player choose.
Both makes sense. If you can talk to them, you can use deception/persuasion like to any other sentient being (and not-really-sentient ones can be manipulated in a similar fashion, they merely need to be able to grasp your words). However, they're still animals. So when trying to get them to do something you are still trying to influence an animal and that expertise will remain helpful.
Just like for contesting grapplings, a player can choose whether to attempt it with strength or dex, I'd rule the same here. If they are attempting something that would reasonably be feasible without talk to animals, they'd get Advantage.
This will be up to your DM
In general, players describe actions and the DM calls for checks. In your case, you've cast speak with animals and are trying to get an animal to do X. You don't define what X is, but it doesn't really matter.
It is then up to the DM to consider what you're trying to do and ask for the roll. There really isn't a "you must use this ability check", it's more about what seems reasonable to the DM. If you don't think it's reasonable, you can ask, but ultimately, it is the DM's call as to what check you make.
I'd give you my ruling, but I don't have enough information. In this case, if I was the DM,I'd want to know what you were trying to convince the animal to do and how you were going to do that. I'd probably do either a deception or persuasion with advantage for the spell assist, but I really do need more info.