If Delta does provide "ground transportation" between the two airports (news to me, and could be useful to others), it could be a ticket on the bus between the two airports (that stops at Jamaica Station, if memory serves me correctly, though it's been several years since I did it, and that was going LGA to JFK), rather than a taxi. Immigration and Customs could, of course, take over an hour. Even with in-cabin (i.e., carry-on) dogs, you'll have to show health certificates, etc. at some point (presumably to Customs). Hopefully that will take just a minute or so extra. You might ask Delta about this or look for info on the Web site for JFK or APHIS (Dept. of Agriculture branch that deals with such issues).

Delta has confirmed that the pets will travel in-cabin on both flights, right? Some arilines limit how many pets can travel in one cabin (I think it's first-reserved, first served).

While hoping for the best, you might do some research on airport-area hotels that accept pets just in case you are delayed more than you anticipate and have to overnight. I've traveled to several countries with cats and know the extra precautions you must be taking, including having all of their health certificates ready to show. However, before you give up on that later flight the same day, you might call Delta again and ask to speak to a supervisor, emphasizing that you're concerned because you are traveling with pets (you can add the in-cabin part if asked, though that could work against your request). The only time I've had the time of a Delta flight changed due to their action, they asked me if the new time would work for me (in my case it meant an even longer connection time between flights so I said no problem).