Michael, my read of that guidance is that the requirement about moving to TLS 1.1 (preferably 1.2) applies beyond the POI context to your internal systems, including (as far as I can tell from what you provided) the links between the servers you talk about. I say that because, first, of the actual PCI requirements that the guidance points to as being implicated here:
Requirement 2.2.3 Implement additional security features for any
required services, protocols, or daemons that are considered to be
Requirement 2.3 Encrypt all non-console administrative
access using strong cryptography.
Requirement 4.1 Use strong cryptography and security protocols to safeguard > sensitive cardholder data during transmission over open, public networks.
Now, merchant-to-you encrypted connections involving POIs are clearly part of this through Req. 4.1. But the reference to Rule 2.3 about encrypting all administrator access stuff and to Rule 2.2.3 with its wider general applicability to many systems that a PCI-covered entity possesses leads me to believe the internal processing connections you're talking about are indeed within the upgrade requirement.
Now, as is usual with PCI stuff one important thing to keep in mind is that there's often quite a lot more flexibility built into the requirements vs. what might seem the case on first blush. For example, the linked-to guidance says that you can meet the "upgrade" mandate if you use encryption in another layer/type that's at least as strong (security-guarantee wise) as TLS 1.1. For example, if you set up a IPsec tunnel between two servers and then run your non-upgradable TLS 1.0 flow through the guidance says that you are & will be in compliance. For more info about flexibilities you might have under the guidance and under the actual Requirements more broadly I'd definitely seek out a PCI consultant or firm that has strong expertise in helping payment gateways & payment processors keep up with new evolutions in PCI rules that are occurring if the wake of the massive thefts of user financial info that have occurred in the last few years. (And, while I'm on the subject of getting direct, reliable advice from an expert, I'll throw in the obligatory disclaimer that I haven't looked extensively into the particulars of your case, may or may not have sufficient expertise--from your standpoint, I'm just a guy on the Internet--to do so, and thus what I have written above is not a substitute for professional counseling.)
In other words, I think your internal networking connections are covered by the upgrade rule, at least to the extent that they convey authentication credential info needed to do administrative things (eg. user names & passwords, of course) and/or carry sensitive card data & other financial info over them. My two cents...
PS: Keep in mind not just the ultimate deadline for upgrading (or making redundant) TLS protections by June of 2016, but the reporting and mitigating measures that the guidance also discusses.