While Grammar Slammer was written with American standards of English in mind, it works for virtually all English including that of the United Kingdom, Ireland, and the former and present British Commonwealth.
Some of our words have different meanings, some are spelled differently, but grammar is the "glue" that holds a language together. Grammar, therefore, is virtually the same wherever English is spoken.
The only significant difference I can think of is that the British sometimes treat collective nouns as plural where the Americans treat them as singular.
UK: "The government are announcing an important decision." (Speaking of specific people in the government).
"The government is a constitutional monarchy." (Speaking of the government as an institution)
US: "The government is announcing an important decision."
"The government is a republic."
I have noticed that some British tend to be more particular about split infinitives and prepositions at the end of a sentence, but those "rules" came from Latin and have little relevance in English. It was, in fact, Sir Winston Churchill who put to rest the "preposition cannot end a sentence rule" with his famous rejoinder that placing the preposition at the end of a sentence is "something up with which we will not put!"
See also Single Quotation Marks
Grammar Slammer does not deal specifically with spelling questions except as they relate to grammar. It does use American spelling in its presentation. Actually the differences between UK and US spelling are also quite trivial.
Norman-derived words that end in -our in the UK end in -or in the USA. Saviour and glamour are preferred spellings in both places.
UK: colour, honour, favouriteSome Norman-derived words that end in -re in the UK end in -er in the US.
US: color, honor, favorite
UK: centre, theatreThe verbs that end in verb-forming suffix -ise in the UK end in -ize in the US. This applies to the few verbs ending in -yse and -yze as well. One exception is chastise which is the same in both places.
US: center, theater
UK: realise, theorise, socialise, analyse,A few other words that end with an s or z sound before a final vowel differ also.
US: realize, theorize, socialize, analyze
UK: cosy, rase (raze is becoming more common), practise (verb; noun is practice)A few words that end in -nse in the U.S. end in -nce in the U.K.:
US: cozy, raze, practice (both noun and verb)
UK: defence, licence (noun; verb is license)A few other words are spelled differently. A few common examples follow.
US: defense, license (noun and verb)
UK: waggon, gaol (jail is becoming more common), mould, moult, manoeuvre, encyclopaedia, furoreThere is not much else that differs. Some differences in doubling of final consonants before adding roots are mentioned in Adding Suffixes Beginning with Vowels.
US: wagon, jail, mold, molt, maneuver, encyclopedia, furor
Pronunciation, of course, varies greatly. There are distinctive patterns of English from around the world. There are also distinctive dialects from different regions--and even neighborhoods--across both the British Isles and North America. People where I live, in the Northeastern part of the U.S., for example, find London English easier to understand than speech from parts of the American South. Pronunciation has nothing to do with grammar. I suspect that someone from East Anglia would find a New England "Yankee" easier to understand than someone from rural Yorkshire or the Hebrides.