The Middle Ages in England lasts for around 400 years (longer, if you include the pre-Conquest era). Up until the 12th century, the main difference in clothing across the classes is in the fabric, dye, and ornamentation rather than in the cut; in fact, the cut for the garments worn by the nobility seems to have been more generous, reflecting the ability to afford more luxury fabric. This is also the case in the 13th century. In the 12th century, while the bliauts of the upper classes did fit tight on the body, they were extremely generous in the skirting and sleeves; while we do not have any extant bliauts to examine, it very much appears that the basics of cut did not change through this period. Your question seems to allude to the later Middle Ages, particularly after the rise of what is often called the "tailoring revolution" in the mid-14th century--which is the move away from clothing cut with simple geometric shapes (squares, triangles, etc.) and towards cuts that are tailored to the body using curved pieces, at least for the outer layers. At the same time, traditional and folk clothing styles continued to use the geometric cuts, which does make them tend to fit differently from the clothing worn by the upper classes--however, they were not actually baggy; they were just not tailored to the body in the same way that garments for the upper classes were. (In fact, baggy garments would have also been wasteful of fabric.) Clothing for peasants remains conservative and evolves much more slowly than that over the upper classes.

The idea that garments using curved pieces cost more is absolutely correct. It's not just in the fabric that is "wasted" in these cuts (it was never completely wasted; the offcuts are known as "cabbage" and were a perk for tailors, who would use the scraps for a variety of purposes), it's in the additional skill required to pattern and sew these garments. The earliest pattern books we have date from the 16th century, and even those are more guides to help the tailor draft and create each individual garment than what we would consider "patterns." Note also that fitting closely on the body or requiring tailoring did not mean that the garments did not require huge amounts of fabric. The later medieval style of the houppelande is a perfect example of this, being a looser outer garment usually worn over more closely fitting inner layers. Changes in fabric production technology (allowing, for instance, for wider widths and the introduction of such fabrics as velvet--a type of silk--for clothing (velvet was initially a fabric used primarily for such items as bedcovers and furniture, contrary to popular depiction; it wasn't even known in England until the 13th century) and the introduction of new dyestuffs also played a part in defining what separated the lower classes from the upper classes.

Stella Mary Newton's work Fashion in the Age of the Black Prince remains one of the defining works about this "tailoring revolution" and is a good place to start if you want to find out more.