If I use a custom DNS like NextDNS instead of my VPN service's DNS (NordVPN), where DNS queries go through the tunnel, will there be a privacy risk or some major disadvantages?
Any perceived benefit you achieve by using a VPN shouldn't be affected by what DNS server you choose, except some may do more privacy-harming things with your data than others. Maybe there will be more latency. Unless you have specific concerns not mentioned in the question, the only true exception to your privacy would be if you somehow chose a DNS server that is locally routable, in which case that traffic would not go through the VPN and you'd have a "DNS leak".
That said, as discussed in the comments, you are simply trusting your data with another company that could be more or less shady than your ISP.