Yes I can see the cat is fine. The recent vaccine guarantees him.
The dog - it depends, getting it revaccinated before you leave guarantees he will be fine as well. A lapse in coverage could do it or not having all the documentation to prove the unbroken chain of vaccinations as listed below:
1). For the EU/UK The first rabies vaccination AFTER microchip implantation is considered a primary vaccination and is only valid for 1 year.
2). The second, if administered within a year from the first, is considered a booster and is valid for 3 years.
All the subsequent vaccinations (if #2 above was done on time) would be considered boosters and valid for 3 years provided they were all done within 3 years of each other. (not 3 years and one day)
That is key. You must have a documented booster within a year of a primary, and then an unbroken trail of boosters all done within the 3 year windows after that. If you have done that AND HAVE ALL the documentation to prove it, then no worries for next year. If you are even late one day you must start over. If you do not have all the documentation to prove it - you must start over. I sense that might be while the vet is saying he needs to be revaccinated. The US standards dont matter. The rules above apply,
If you do not have all of the documentation to show that she received her first booster within the 1 year period - then All of the subsequent 3 year vaccines would be each be considered primary for UK/EU purposes regardless of it being ok in the US.