A man who married a British woman would be entitled to settle in the UK and his children would be UK citizens, and thus entitled to British passports, but he himself would not, unless he was naturalised. Passport applications are no longer archived, alas. The National Archives has indexes of applications up until 1948, with those from 1851 to 1903 available online via FindmyPast, but these contain only the name of the applicant and the passport number, so are of practically no genealogical value.

As I said, he would only have been able to apply for a British passport if he had undergone naturalisation, so you may wish to try searching for records of this. The National Archives guide may help www.nationalarchives.gov.uk/records/research-guides/naturali....