The Great Vowel Shift
There is a very considerable literature on this change or series of related changes (see Lass 1989, 1992, Stockwell and Minkova 1990, Johnston 1992). However, the main point is the identification of some discrepancies in the application of the Great Vowel Shift in the North and South, and the development of an input system for the SVLR. I therefore only list the stages of what is conventionally known as the GVS below in roughly chronological order, largely following Johnston (1980).
Stage 1 (c.1400 - 1450 North, 1450 - 1500 South)

In this earliest subshift, high-mid front (and, in the South, back) long monophthongs raised, while originally high long vowels diphthongized (1). The failure of /u:/ to diphthongize in the North, leading to Modern Scots /ku/ cow and /hus/ house versus RP /kaʊ/, /haʊs/, may be ascribed to earlier /o:/-Fronting; since there is no /o:/ to raise in the North, there is no pressure on /u:/ to diphthongize. SSE typically has diphthongal [kΛu] or [ka:u], the realization varying speaker-specifically.
(1)

The high vowels are shown as shifting to some intermediate value, rather than directly to [ai] and [au], because an immediate full shift would counter-historically merge /i:/ with /ai/ and /u:/ with /au/ (Johnston 1997a). In other words, lexical items with original Middle English /i: u:/ and /ai au/ surface in Modern English with different vowels, so that /i:/ may not shift to /ai/, nor /u:/ to /au/, until /ai/ and /au/ have in turn moved away from these values; whether or not we regard avoidance of merger as a general linguistic tendency (see Lass 1976), it is clear empirically that no merger took place in this case. The precise identity of the intermediate values is much debated, and several options are given in (1). Although [əi əu] are frequently suggested, Lass (1989) argues against early centralization on orthoepical grounds, pointing out that Hart's (1569) representations of reid `ride' and hound `hound' signal two unrounded front elements in the former, and two back rounded elements in the latter: indeed, `no orthoepist before Hodges (1644) appears to report anything that could be construed as a central vowel in the relevant position' (Lass 1989: 91). Regardless of precise realization, this inter mediate stage of Vowel Shift for the original high vowels will be extremely important for the analysis of SVLR developed below.

Stage 2 (c.1450 - 1500 North, 1550 - 1620 South)
Whereas Stage 1 of the GVS affected the Northern and Southern vowel systems rather differently, Stage 2 produced the same results in both areas (2).
(2) 

There is one discrepancy, resulting from the earlier monophthongization of final /ai/ in Scots and the North in frequently occurring lexical items; this /a:/ raises regularly to /ε:/ in Stage 2 of the Vowel Shift (and subsequently to /e:/). However, in certain dialects of Scots, /a:/ failed to raise when preceded by a labial consonant. In this case, /a:/ might be retained, or, in other areas, the influence of the adjacent labial appears to have caused a post-GVS backing and rounding of /a:/ to /ɔ:/. These changes result in the various possible pronunciations of two in Scots.
Stage 3 (c.1490 - 1510 North, 1600 - 1630 South)
(3) 

In Stage 3 of the GVS, the subshift of /au/ to /ɒ:/ again involves an intermediate, partial shift, since a direct movement of /au/ /ɔ:/ would be ordered before the raising of /ɔ:/ /o:/ in the final stage of the GVS, and the merger which would result is not attested: compare Modern Scots law, cause, saw, where /au/ /ɔ/ (or /a/), with throat, coal, where the /ɔ:/ /o:/ shift took place.

Two discrepancies between North and South are relevant to this stage of the GVS. First, whereas in the South all /εi/ ( /ai/) monophthongized to /ε:/, in the North this development took place only medially. Where /εi/ occurred word-finally in the North (and, it will be recalled, this will be the case only in relatively uncommon words, since final /ai/ in frequently occurring words had earlier monophthongized to /a:/), it remained diphthongal and developed to Modern Scots /Λi/. This accounts for the differing pronunciations, in some varieties of Modern Scots, of pail, pair, rain with earlier medial /εi/ and thus modern /e/, and pay, way, with final /εi/ /ai/, and modern /Λi/. Again, this is of direct relevance to SVLR. Second, /ɔu/ monophthongized to /ɔ:/ only in the South, raising in the final stage of GVS to /o:/ and subsequently diphthongizing to give RP /oʊ/. In the North, however, /ɔu/ is retained and later becomes /Λu/, as in grow, [grΛu].

Stage 4 (c.1500 - 1550 North, 1690 - 1715 South)
In this final complex of shifts, the mid-front and low-mid back monophthongs raised, with /ɒ:/ becoming /ɔ:/, while the first element of the /eu/ diphthong also raised, giving /iu/ (see (4)).
(4) 
 

After the completion of the GVS, /ʊ/ lowered to /Λ/. This lowering was complete in Scots, partial in the South, and failed in the North of England, producing the present-day division of dialects with only /Λ/, both /ʊ/ and /Λ/, or only /ʊ/.

The Scots vowel system after the Great Vowel Shift is shown in (5).
(5) 

Note that /ε:/ has been lost completely through the operation of GVS, while /a:/ and /ɔ:/ are fairly marginal; /a:/ occurs only in certain dialects for earlier /a:/ (including /a:/ /ai/) after a labial consonant, as in two [twa:], away [əwa:], while /ɔ:/ has the same origin in a different set of dialects, in which earlier /a:/ backed and rounded under the influence of a preceding labial ± so [twɔ:], [əwɔ:]. The monophthongization of earlier /au/ to either /ɔ:/ or /a:/ in words like law, craw `crow' gives one additional source for these vowels. A system of the type shown in (5), with a few revisions to be discussed below, would have formed the input for SVLR.