Forgive the levity; that pooling water on the side of your prospective residence is unamusing, I'm sure. Firstly, kudos for doing the kind of detective work so many buyers fail to do. Secondly, I'm curious what the sellers and their agent have told you about the issue because it's really up to that party to address the soggy situation -- and disclose it on the contract as a water-infiltration problem. If they aren't disclosing this, you should be wondering what else they aren't divulging.
Whether the neighbor's apparent water-diversion strategy is legal is subject to debate. The laws on this, like that yard, are a little muddy. Sometimes it takes a city inspector's assessment or a threat of a lawsuit (or both) to get things moving in the proper direction. As a rule of thumb, though, neighbors aren't liable for harm caused by the naturally occurring lay of their land. But if they alter their property in a way that intentionally forces water onto the next-door neighbor's property, the court might find them in violation of the so-called reasonableness rule, which is defined on a case-by-case basis.
In your case, for example, if the neighbor changed the drainage flow with new gutters and had his contractor design the poolside deck to drain toward your prospective home's yard, a court may find that the resultant flooding was foreseeable. However, the complainant in most jurisdictions typically has to prove both damage and nuisance. In your case, don't hold your breath waiting for the soon-to-move seller to take legal action.
If the sellers won't rectify the situation but you're still enamored by the place and want to buy it anyway, make sure your home inspector carefully assesses just how serious the drainage problem is. If there's some question, you may need to call a seasoned landscaper out to look at the problem and recommend remedial measures. It might be that a berm, gravel area (French drain), catch basin or retaining wall can be installed to mitigate the flow and help direct it to a natural drainage easement that will send the water to a public area. It also might be that a there's a natural spring under the property that's exacerbating the situation.
Regardless, get some cost estimates, and don't hesitate to ask for a credit from the sellers to help with all or part of this. Of course, the sellers may just opt to sell to a less picky (read "less discriminating") buyer, especially if they're selling as is and have other buyers on the hook. There's also a chance, I might add, that your lender may make your loan contingent on fixing the problem if the bank's appraisal uncovers the drainage issue -- unless, that is, you set up a "repair escrow" fund to fix the problem post-closing.
However you proceed, know that perennially pooling water can have a destructive long-term impact on a home in the form of wood rot, fungus, soil runoff and soil instability. So unless there's a simple and relatively inexpensive solution to the drainage issue, you just might have to find a different property that floats your boat -- figuratively this time! Good luck.