Regrets American uncle turning up in rubber (7)
I believe the answer is:
masseur
'rubber' is the definition.
(someone who rubs or massages people)
'regrets american uncle turning up' is the wordplay.
'regrets' becomes 'rues' ('rue' can be a synonym of 'regret').
'american uncle' becomes 'Sam' (Uncle Sam is a nickname for the USA).
'turning up' says the letters should be written in reverse.
'rues'+'sam'='ruessam'
'ruessam' reversed gives 'MASSEUR'.
'in' acts as a link.
(Other definitions for masseur that I've seen before include "Muscle manipulator" , "One who kneads and strokes" , "He kneads the body" , "Physiotherapist" , "One kneading joints" .)