Once you have found an individual in the 1939 Register it is possible, with some effort, to find their exact date of birth as recorded in the Register and transcribed by FindMyPast without 'unlocking' the household.
The process requires you to use the Advanced Search option. Using the exact year of birth (-/+ 0yrs) from the preview result, try each month (1 through 12) until you find the same person again. Then repeat for each day of that month until you find them again. You now have the year, month and day of their birth. You might find it also helps to include the Borough for a better result.