There are people who actually can't pronounce the 'r' sound properly. A famous example being Jonathon Ross. There are also people who lisp and make a 'th' sound instead of an 's'. I don't know what the cause is but I doubt very much if the incidence is any less in America.
However, an uninformed American coming to Britain may think people who have NO problem still 'mispronounce' their 'r's depending on what area he/she visits. Why? Because some English accents are rhotic, like American English and some are not.
Rhotic speakers pronounce the 'r' strongly , it is especially noticeable on word endings such as 'father' mother' and of course 'cider'. Other English accents are non-rhotic. The 'r' isn't pronounced and words such as 'father' 'mother' etc end in a schwa, represented by the phonetic symbol /ə/.
In a non-rhotic English accent a word like 'hard' is pronounced /ha:d/ and 'father' is pronounced /Fɑːðə/
Most accents in England are non-rhotic. However , despite how it may sound to an American, it is not a 'mispronunciation' but a perfectly normal phonetic feature of British English.
