I think if anything, networking, load balancing, and firewalls will always be the biggest effort in any VDI implementation. The number of cogs that must spin just right is no small number.
That said, with a little insight, you can break right through the fog your dealing with.
Based on your description, there can be a few causes and steps to troubleshoot. One item to mention when discussing issues with UAGs is the number of NICs for that UAG. Adding two or more NICs is fine, just note it adds one more layer which must be done correctly.
Possible Causes
1. External UAG URLs are not matching
2. The UAG IPs or Subnet do not have access to the VDI VLAN for required ports
3. With Multiple UAG NICs, static routing is easy to misconfigure.
4. Possible Load Balance VIP group configuration issue
Research Steps
1. Review the client connections networking requirements in this excellent Technet Article - Link.
1.a - That article will show you in details the source and destination for every horizon component.
2. Review Troubleshooting Firewall & Connection Issues for the UAG on this VMware Doc - Link
3. include getting familiar with manager log on UAG and Connection Server logs to correlate where in the path to the desktop the failure occurred.
Investigation Steps
1. First, verify that both UAGs are green status on Horizon Admin Console
2. Externally, use telnet or similar tool to verify each port is open.
3. SSH into UAG, verify required ports are open to load balance VIP, connection server, and VDI desktop IP
4. Leverage TCP Dump steps in the UAG troubleshooting, to watch live attempts
5. Finally, simplify approach by first leaving only one member in the LB VIP group. Then, remove the LB VIP group, NAT directly to UAG, etc, etc. Keep simplifying until root cause is found.
Finally, try to do as many of these steps without contacting the networking team. When building a good relationship with other IT teams, I have found doing my homework as much as possible and bringing solid evidence to be worth its weight in gold.
Good Luck!