Under the 1917 Royal Warrant, children of a monarch, male line grandchildren and the eldest son of the eldest son of the Prince of Wales are HRH Prince/Princess from birth. Presumably this will be extended to cover a female heir apparent under the succession to the Crown Bill currently going through Parliament.
Whether the Earl of Strathearn's eldest son has a specific Scottish title depends on whether or not the Duke of Rothesay has a third Scottish title. Normal practice with peerages is that the eldest son uses his father's second title by courtesy (i.e. he is called by it), and the eldest son's eldest son uses the third, provided, of course, that there is one. There were problems in 1914 when Arthur of Connaught's son (the future second Duke of Connaught) was born, as no one realised that, following practice up to then, he should have been HH Prince Alasdair of Connaught, and the Duke of Connaught didn't have a third peerage. Things then drifted and the young man ended up using his mother's second title (she was Duchess of Fife in her own right), and was known as the Earl of MacDuff until he succeeded his grandfather in 1942.
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