If you haven't already placed a dehumidifier in your Mom's basement, you should seriously consider doing that. . . Make sure to size the dehumidifier to the square footage of the basement. It won't remove any existing mold, but it should stunt the growth of any new mold from forming as long as you keep it running to maintain a humidity level around 50-60%. Mold only grows if an environment is humid enough to support it's growth. Quality dehumidifiers are designed to run continuously, if needed. I recommend Whirlpool dehumidifiers due to personal experience w/ that brand in my wife's dog grooming shop that had an awful humidity problem, including mold on the walls when she purchased the business. It's internal timer turns it on whenever the humidity level gets to 50%, and you'd be amazed at how many gallons of water it sucks out of the air every day. She no longer has any mold on any of the surfaces. (I'm in the Atlanta area, and have a similar climate as you have in DC in re: to humidity) Whirlpool dehumidifiers can be found on Amazon at very competitive prices, and w/ free shipping. Also, Whirlpool is the only manufacturer that offers a 5 yr warranty on the sealed system in their dehumidifiers; most all other dehumidifiers come w/ only a 1 yr warranty. (note: I do not have any affiliation w/ Whirlpool, Amazon, or anyone except my wife - lol)
Assuming that the heating system's air ducts are in the basement, seal up any leaks in the ductwork, as introducing heated air into that space would most likely increase the humidity. Use aluminized tape, and not duct tape, to seal any leaks. Common gray duct tape is not for sealing duct work, despite what it's usually labeled. . .
To remove mold - esp. black mold - can be hazardous, and is best left up to professionals. . .Unfortunately, that's a very costly proposition, as it usually involves the crew having to wear specialized protective clothing, air-extraction of the affected space, tenting of the entire structure, disposal of most floor & window coverings, and proper disposal in monitored hazardous waste facilities, etc. That work can easily cost in the several thousands of $$s, depending on the affected space. (beware of mold-remediation contractors, as the industry has it's fair share of scammers) Mold can accumulate on exposed wood framing, such as overhead floor joists and the underside of sub-flooring, on ductwork, pipes, walls, and most anything in a basement. During removal it should not be allowed to become airborne. There's another way for a homeowner to deal w/ removing mold, and that's by encapsulating it by over-coating & thus, sealing it, w/ the appropriate type of paint, after killing any existing mold w/ a solution of bleach or other specialized products designed for that purpose. Consult w/ a major name brand paint retailer, seek advice from experienced house painters, and do some research online to determine if that is something that you want to tackle by yourself. Just be careful to wear protective clothing, gloves, a hat, eye protection, and a respirator. Prepare to responsibly dispose of any clothing you had worn after doing the mold removal.
Another important consideration is that when one discovers mold in their property, it becomes that person's liability to have it removed. Due to Federal and other municipality laws, the liability can not be passed on to a new owner when the property is sold, so it's best to deal w/ remediating it as soon as possible so that it doesn't get worse than it already is. . .Keep this in mind when you're discussing this situation w/ your insurance company (as they might cancel your coverage), your neighbors, and esp if you should involve the media, as it could adversely affect the resale of the property now, and at a later date. All the more reason to discuss options w/ an attorney (as much as most of us hate to), since it seems evident from what you've written that your Mom's property (and it's contents) have suffered significant & on-going damage caused by your neighbors. The neighbor(s)' homeowners insurance policy might very likely be liable for damages that their insured's actions have caused your family, whether or not the neighbor(s) violated any building codes such as set-backs, not securing permits, etc. Take pictures, keep dated records of any damage (esp when the flooding scenario first occurred, coinciding w/ when the neighbors altered their structures, backyard, and so on), and construct a record of any verbal discussions & written correspondence between members of your household and the offending neighbors.
I wouldn't be at at all intimidated by the neighbor who is a building inspector, as his career could be in jeopardy if he's violated any of the laws that he's employed to enforce.
Like someone else wrote: "building inspectors have bosses", and they won't want adverse publicity concerning their department's personnel, and esp if they ignored the issue, as it's also would involve the health department. . .To ignore the mold is going to affect the well-being of your Mom, and anyone else in the home, including pets. PM me if you need any more info, as I've had to deal w/ this subject on more than several occasions as a Realtor, as a contractor, & a homeowner.