Solid lawful advice.Hate to say it but your first move would probably be consulting with a licensed engineer that’s experienced in such things in your municipality. Anything that discharges off your property can draw the attention of the county.
If there’s any option of restoring the original drainage underneath your neighbors wall that may be the best bet. Perhaps you can work with them to improve their drainage at the same time.
Our soil is pretty sandy and drains fast. My initial thought was a drywell/basin alone without a pump. But a little math on roof and driveway runoff area convinced me that I'd be a fool not to plan for a pump.I guess it all depends on the kind of ground you have. Where I live there is no such thing as truly flat ground. We have sand and clay. When dry it’s almost waterproof but once the powder is soaked it’s a total sponge.
If you can do it cheap, and on the weekend/night go for it.Solid lawful advice.
Restoring the original drainage would be more trouble than it's worth in this case. Between the concrete driveway, block wall and its concrete footer(!) there is nowhere to dig that doesn't involve costly new concrete work. The driveway's houseward side is nearly as tight. But without a wall footer on that side, I did have room to bury some sprinkler supply lines years ago. There's room enough to re-excavate and add another line out to the street.
I care about doing things "right" but also sensibly. I meet or exceed state and local building code whenever possible. But I'm also willing take the risk/responsibility of respecting the "spirt" of the code where city planners are more concerned with right than sensible. I think there's an under $1000 solution to this, with good planning and a lot of digging. A $10,000+ hardscape solution is just not in the cards.