The potential issue in this is not in any rule mechanics, but in the cleric's play-balance.
By the question "what aspects might go awry?" I am going to assume you mean "what issues will I have to manage either in setting up a character or in what might happen during play from the House Rule that clerics can gain their divine powers from a non-deity source?". I'm going to refer to the source of power for the character you have described as "the weather".
I believe that very little, if anything, is needed in terms of the mechanics of the rules as you would be treating "the weather" as if it were a deity. An obvious example of this working is the Druid, where the divine power source can be "the force of nature".
Power of Nature (PHB p.64):
Druids revere nature above all, gaining their spells and other magical powers either from the force of nature itself or from a nature deity.
Mechanically this concept would work for all the cleric's powers, even Divine Intervention (a discussion point in other answers) as it is down to the DM what actually happens:
Divine Intervention (PHB p.59)
The DM chooses the nature of the intervention; the effect of any cleric spell or cleric domain spell would be appropriate.
I think the real issue to consider is a cleric's core class concept of being a divine agent, granted power to further the cause of their deity in the mortal world. The cost of the divine power bestowed upon them is to be devoted to the deity's cause and principles, manifesting in the role playing of the character. This is actually a really important part of the cleric's play balance as it gives the DM a clear and rules based mandate to sanction a cleric for "bad" behaviour and ultimately strip them of their divine abilities if the character continues to defy their deity. It means a cleric character can't act how it wants whenever it wants, the powers come at a cost.
A house rule to allow a source of divine power with no such agenda effectively removes this core restriction and could significantly effect game balance, as the character is granted it's divine powers "for free" compared to other clerics. It occurs to me that this character would effectively be a divine warlock using the cleric class for the rules.
Whether this is an issue is down to how you run your game, how much your player will exploit the house rule and whether your troupe is comfortable with the play balance of the character. You may also need to consider what happens when the next player comes up with a character using the house rule that effects play balance even more?
An obvious option to manage this would be to give "the weather" an agenda and the will to impose this agenda on its clerics, but that makes it a deity in every way that matters and makes this question void.
I think it's an interesting character concept. From a DM's perspective I would make it clear that while "the weather" is not a deity as such, the character will have requirements on it's behaviour in the same way that "the force of nature" does on a Druid, just as inconvenient as if they worshiped a deity.