B757 Captain reported full lavatory waste tanks during cruise on an international flight with one lavatory inoperative for dispatch. The flight diverted to an alternate airport for lavatory servicing where maintenance determined the lavatories were not previously serviced prior to departure.

Narrative

We left ZZZZ with the mid-lav inop. The proper calculations were made in regards to the number of paxs with 1-lav inop. My FO checked the lavs as best as a person can for the 757. We also received the notification that the lavs had been serviced in ZZZZ.2-hours into the flight; the fwd lav was already full. The decision to restrict first class paxs only to use this lav was made. All other would have to use the aft lavs. The FA in the back advised me that there was a steady line of paxs for the aft lavs for over an hour. A little while later; the aft FA advised me that both aft lavs were now full. We still had 4-hours left in the flight back to ZZZ.I had a satcom call with dispatch. He was not very helpful at all. We spoke with Maintenance Control and he confirmed the calculations were done correctly. No more help needed from Mx (Maintenance).We then asked dispatch for his opinion and he asked if we were thinking diversion. We said yes. His immediate suggestion was ZZZZ1. Yes it was closer than ZZZ1; however; he did not take into account customs; pax; and getting the services we needed.My FO and I wanted to divert to ZZZ1 because it is a large station for us; can comfortably accommodate our aircraft; and has US customs if needed. Dispatch agreed. Once again; dispatch was of little help.We asked dispatch for help to get us routing to ZZZ1. He said he couldn't and that we would have to do all of that coordination. Once again; dispatch was of little help. Me and my FO made contact with ATC and secured a new clearance to land in ZZZ1.Made it to ZZZ1 uneventfully and parked remotely. Mx immediately boarded the aircraft and confirmed that the mid-lav was labeled correctly as inoperable. Mx then informed us that the lavs had NOT been dumped in ZZZZ and that is why our lav tanks filled up so quickly. Ugh!!We spoke with the Chief Pilot and it was determined that we were not legal to proceed to DH (Dead Head) to ZZZ. Dispatch was very adamant that we were and to just stay on the plane and it will all work out. Nope!We cleared customs and then called the Dispatcher. They advised us that we were to DH to ZZZ that night. I spoke with her and advised her that I had spoken with the Chief Pilot; he had spoken with another Dispatcher and also the operations manager and we would need a hotel for the night. After a 15-minute wait; our pairing was updated and we were given a hotel for the night.We diverted because the lavs were full up to the flapper and were worried about a haz mat situation.Throughout this entire situation; we felt like there was little to no help from the dispatcher and that we were left on our own to accomplish a lot of things. He just kept saying it was our responsibility to do it. Dispatch was definitely not 1-stop shopping. It was also very challenging to communicate with dispatch because of a language barrier. His accent was so thick that we struggled to understand what was being said.