Your uncle may not have known, and probably would have continued to call himself UDT, even if he had, my guess is that few of the old UDT call themselves SEALs, but UDT was merged into the SEALs about 1983, with the UDT desiring to stay in, receiving additional training.
My uncle’s wife said he was proud of his service, didn’t care to be known. He knew he did his duty.