I'm afraid that the basic premise of the question is incorrect. The creators of the Cyrillic alphabet and, most likely Kliment of Ochrid or Konstantin of Preslav, much like their predecessors, St. Cyril and Methodius (considered to be authors of Glagolitic alphabet and Old Church Slavonic language) most certainly knew of the existence of Latin alphabet and it is possible they knew Latin of maybe even some Western language. But the Cyrillic alphabet is not a derivative of the Latin one, beacuse its creators simply used the set of letters they intimately knew and that was used for everyday writing in the area they hailed from. And this was, of course, Greek alphabet.

Any relation to English is, of course irrelevant here, as the first form of Cyrillic has been created in the first half of the 9th century, when English has not existed yet, having only begun its shift from distinctly Germanic Old English to Middle English, and we can say with almost complete certainty, that the clergymen living in the northwestern part of the Eastern Roman Empire did not know any of these languages.

This alone provides the explanation of the shape of some letters right off the bat. Most notably, the letter ρ is not Latin 'p' with changed phonetic value, but a Greek letter 'rho', phonetic value of which is equivalent to Latin 'r'. Likewise, 'χ' is not 'x' but 'chi', with a phonetic value similar to 'h' in English word 'hold'. Cyrillic 'Υ' is derived from 'ypsilon' that in Eastern Roman Empire has been though to be pronounced as a vowel between Slavic 'u' and 'y', much like 'u' in French word 'lune' or German 'ü', hence its association with the East Slavic open 'u' (in Slavic languages it is not ioted). Cyrillic 'З' is not an inverted 'E', but a a sign derived from Greek 'dzeta' (ζ). Cyrillic 'ϲ' is not what Latin renders in English or German as 'ϲ' (in Western Slavic languages it has a completely different phonetic value, an equivalent to Cyrillic 'ц'), but a Greek ending 'sigma' ('ς' or 'ϲ'), especially its latter, older shape. Other Cyrillic letters also present their origins quite clearly, especially in the case of 'п' (Greek 'π'), 'ф' (Greek 'φ'), 'д' (Greek 'Δ'), 'л' (Greek 'λ') etc.

Completely new letters, not present in Greek or Latin alphabet were created for the sounds that were not present in Greek and thus had to be rendered by new symbols. This is how we got 'ц' (ts) all the voiceless postalveolar affricate and fricatives 'ч', (ch) 'ж' (zh), 'ш' (sh) and 'щ' (shch), with the latter two being distinctly similar to Phoenician and Hebrew 'shin' with a phonetic value equivalent to English 'sh'. Letter 'ж' is also likely to be influenced by the Glagolitic counterpart (Ⰶ).

Now for something more complex. The Cyrillic ioted consonants (е [ye], ё [yo] я [ya] and ю [yu]) were initially written with as a digraph created of ligatured iota and a respective vowel. for example, an early forms of я and е was written as 'ꙗ' and 'ѥ' respectively, while 'ю' is derived from merging of 'ι' and letter 'ѹ' (that later was abandoned), where you can clearly see the leading iota. In the course of centuries, 'ꙗ' has been simplified into a sign resembling inverted Latin capital 'R', although the similarity is purely incidental (much like that of Latin cursive 'n' and Hebrew 'chet' with the phonetic value of pharyngeal 'h').

In addition, the letter 'beta' (β) has been used for the phonetic value of Latin 'v', because in late 8th and early 9th century, it has already been used in that capacity in Greek and thus it was natural for the native Greek users to associate former 'beta' with sound 'v', especially given that in many languages it can be phonetically rendered as a bilabial approximant or labiodental approximant depending on context or dialect (it is quite notable in Spanish, where the sound like in e.g. the word 'lavar' sounds to a Slavic or English ear like something between 'b' and 'v'.

So, to sum it up, the apparent phonetic differences of the Latin and Russian script (that is more correctly referred to as 'grazhdanka' or 'citizen script' after the reforms of Peter I around 1708) are a result of the Cyrillic alphabet, now used in many Eastern European countries being based on Greek, not Latin alphabet and thus retains the many traits of the former.