Is there any significance to the pattern we get when the Roman alphabet (upon which English is based) is arranged by giving vowels a "lead" column (which I hope you will be able to see as a grid)?
No, this pattern is an accident. It developed from ancient alphabets which had a similar order but no similar pattern because their collection of letters was different.
For all the adaptations and mutations, the alphabet's order of letters has been relatively stable. In the 1920s, archaeologists found a dozen stone tablets used in a school in Ugarit, a city in what is now Syria, that are from the fourteenth century BC and preserve two orders of the Ugaritic alphabet. One, the "Northern Semitic order" is related to the Phoenician and Hebrew alphabets and features bits and pieces of an order familiar to Modern English speakers: a, b…g, h…l, m…q,r.