According to research by Dr Howard Colvin, described by Lord Aberdare (The Willis Faber Book of Tennis & Rackets, Quiller Press, 1998), the original wooden court at Hampton Court was built before 1532. The present court there was built on the site of this wooden court by Charles I in about 1625. 

And:
Roger Morgan writes: "The oldest court still standing is that at Falkland Palace which was built in 1539. . . . The tennis court at Hampton Court Palace is the next in terms of age which still exists, and was built in 1625." page 98 in Tennis: The Development of the European Ball Game, Ronaldson Publications, Oxford, 1995.

And:
Kathryn McNicoll (sister of former World Champion Chris Ronaldson) in her excellent booklet Real Tennis (Shire, 2005) agrees that Falkland has "...the oldest court of all...".

However:
Surprisingly, Heiner Gillmeister disagrees, and states that the oldest court in use is at Hampton Court (Tennis A Cultural History, Ronaldson Publications, Oxford & Leicester University Press, 1997). Needless to say that Hampton Court's website makes the same claim.

So:
It seems likely that the disagreement over the oldest court is due to the fact that the (now demolished) 1532 court at Hampton Court was older than the court at Falkland Palace (1539). The present court at Hampton Court was not built until 1625.

Finally:
The Royal Tennis Court at Hampton Court Palace, where tennis has been played almost uninterrupted since King Henry VIII first built a court there, is regarded as the world's centre or "home" of real tennis, in the same way as St Andrews is regarded as the "home" of golf. (Tennis was not first played at Hampton Court, nor was golf first played at St Andrews).