My background is much more tied to early modern Scotland, but I think I can at least shed some light that may spill over across the Tweed.

What I think that 'fundamental law' is referring to here is the 'package,' if you will, of precedents that govern the conventions of rulership and the relationship between king and Parliament. It was used pretty aggressively in the seventeenth century to justify Parliamentary resistance against unpopular royal policies like Ship Money, etc (I imagine this is the context in which you're encountering this, though the 1688 Glorious Revolution probably hits a lot of the same legal/constitutional buttons). Things like Magna Carta, the 1628 Petition of Right, etc. The 18th-century judge William Blackstone wrote an influential legal commentary that it may behoove you to take a look at if you're curious. I'll also quote from J.G.A Pocock's The Ancient Constitution and the Feudal Law:

. . . by the time of the Apology of 1604 the Commons were already insisting that the whole body of their privileges should be recognized as theirs by right of time immemorial. The search for precedents resulted in the building-up of a body of alleged rights and privileges that were supposed to be immemorial, and this, coupled with the general belief that England was ruled by law and that this law was itself immemorial, resulted in turn in that most important and elusive of seventeenth-century concepts, the fundamental law. . . . [I]f you had asked a seventeenth-century Englishman the question 'What is it that makes the fundamental law fundamental?' he might indeed have been embarrassed for an answer, but would probably in the end have replied: 'Its antiquity, its character as the immemorial custom of England.' . . . The fundamental law or constitution was an ancient law or constitution; the concept had been built up by the search for precedents coupled with the common-law habit of mind that made it fatally easy to presume that anything which was in the common law, and which it was desired to emphasize, was immemorial. [48-49, 1987 edition. Suggested further reading includes J W Gough, Fundamental Law in English Constitutional History]

As for how this spills over into politics, I'll have to answer this from a Scottish perspective (my apologies). Scotland, as you're probably aware, uses a different sort of law than England, and Scotland's conception of its own 'ancient constitution' is much more hazily defined than in England's case, and doesn't refer to specific statutes, charters, or laws as often. It's instead tied to a fundamental conception of the Scottish king as the centrepiece of Scottish nationhood, so take everything I say in this part of my response with a grain of salt in terms of its direct translation to an English context.

So, if you study 17th century England, you've probably heard of the Covenanters, the rascally Scots who got mad at Charles for fiddling with the Jacobean church settlement in Scotland and started the Bishops' Wars. The situation there is actually much more complicated (that explanation is like saying that the English Civil War was fought over unpopular taxes: it's true only at a very, very broad level). Basically, over the course of the late medieval and early modern period, you have an increasingly popular historical conception of the Scottish 'ancient constitution' as more 'democratic' (or at least aristocratic) than it probably ever was.

The first person I've read who really did this was John Mair (or Major), an early sixteenth century theologian and historian who taught John Calvin, among others. A quick side note, some earlier texts like the Liber Pluscardensis and Walter Bower's Scotichronicon arguably fall into this tradition, but I'm not as familiar with them. Mair came from a Catholic conciliarist background, and his ecclesiastical political stances spilled over into his secular history: he argued in A History of Greater Britain (1521) that the king's authority was fundamentally ministerial instead of absolute (that is to say, he ruled the kingdom as an administrator for the common good rather than as a personal plaything granted by God), and that his office had been instituted by the people (not by divine right) at the legendary beginning of the Scottish monarchy in 330 BCE. Here's a link to the book.

The next person who contributed to this aristocratic-constitutional conception of monarchy was Hector Boece, a humanist scholar and theologian who wrote his History of the Scots in 1526. This book made similar arguments, but presented them in a way that imitated the medieval speculum principis genre of literature (mirror of the prince, basically handbooks for rulership, often used illustrative examples). His book basically presented a long list of kings, with good kings who listen to their counselors and heed their advice, who govern by consensus etc. (you get the picture). On the flip side, he presents a long list of bad kings who ignore their counselors, don't care about the public good, who oppress the people with their greed or lechery, etc. The point he's essentially making is that the good kings of Scotland, from time immemorial, have adhered to a certain code of conduct and ruled with the advice and consent of their nobles. Here's his book.

The last person to really contribute to this historic-constitutional tradition of Scottish historiography that I'm going to mention is George Buchanan. He was an internationally renowned humanist scholar, and also a rabid opponent of Queen Mary. He wrote his History of Scotland and his Dialogue on the Law of Kingship Among the Scots in response to her deposition, in order to justify it. Basically, the point he's making is that Scots made their own king and granted him ministerial authority by consent, and that he had to rule with the Scottish Estates (Parliament).

What this has to do with the 17th century Wars of the Three Kingdoms is that these contributed to and (in my view) decisively shaped a very popular understanding of Scottish history, and by extension, how the Scottish kingdom should be run. When King Charles instituted policies like the 1625 Act of Revocation in Scotland, or the later attempts to anglicize the Church of Scotland based on royal prerogative, a healthy contingent of Scots joined the Covenanters in 1638 to defend the "ancient constitution" of Scotland. Many of the nobles and moderate ministers who joined the Covenanters did not understand their resistance as an attempt to totally reshape society into a theocratic form (like more radical Scots such as Samuel Rutherford or Archibald Johnston of Wariston). In England, you have a similar phenomenon of an increasingly aristocratic/'democratic' view of the English constitution and skepticism towards royal authority (though, once again, I may be missing a nuance here - England is not my specialty). This resistance to royal centralization (Strafford and Laud's 'thorough' policies) didn't take the form of a stand for universal principles of democracy or human brotherhood at first (and only some like the Diggers took those up later): the defense of 'fundamental law' was a defense of traditional privilege on the part of magnates and gentry who felt their rights were being intruded upon by a grasping, centralizing monarchy. Appeals to 'fundamental law' dress this defense of privilege into a defense of the kingdom as a whole (and many no doubt sincerely saw it that way; I am in no way casting judgment on the English Parliamentarians for exercising their God-given right to rational self-interest).

As for your last question, I don't think it's separate at all from parliamentary sovereignty, and in fact I think that the idea was forming that parliamentary sovereignty was the fundamental law of England.

Hope this helps! Sorry for the lengthy Scottish digression, but it's what I know. No one ever asks about early modern political thought so I felt the need to jump in and hammer this novel out.

edit: Wow four pages when formatted like an essay into Word. I really should have just worked on my thesis instead.