I've been looking into naval history recently and I've seen that although examples of light cruisers (HMS Belfast, HMS Caroline) and destroyers (HMS Cavalier) have been kept as museum ships, there are no preserved Royal Navy battleships or battlecruisers.
I would have thought that at least HMS Dreadnought may have been preserved due to its historical significance as having changed naval warfare for the first half of the 20th century.
So why is this? Why were some of the lighter warships sold to trusts for preservation whilst the larger ones were not?
The MoD has announced that the 22000 ton aircraft carrier "[HMS Illustrious](https://en.wikipedia.org/wiki/HMS_Illustrious_(R06))" (launched in 1978) will be preserved for the nation. This might fit your definition of big 20th century royal navy warship.
Holland1 (the first submarine of the Royal Navy) too is on display after being recovered.
It is true that some of the most iconic large naval ships have not been preserved. HMS Dreadnought was scrapped for GBP 44000; HMS Hood was sunk; HMS Hermes sold to India.
The practice of setting up trusts for ships as maritime museums was much less prevalent during the early 20th century and money from scrap or for preservation (large battleships aren't cheap to maintain) much more a focus. The practice extends to other large and less iconic ships too. However someone more familiar with the specifics of the sunsetting of these battleships may be able to answer better.
There is still hope that Britain's first nuclear submarine will be preserved, so you may still be able to visit HMS Dreadnought as a museum ship! (even if it isn't the iconic battleship)
OK. Some answers on why no big ship from a certain portion of the first half of the 20th century would be preserved.
HMS Dreadnought was sold for scrap on May 1921 and finally broken upon arrival at the breakers on 2 Jan 1923
The Washington naval treaty (negotiated between Nov 1921 - Feb 1922) set hard limits on capital ships including total tonnage by country; the text even defined which ships could be retained by which country and how other ships were to be disposed off (scrapping, sinking, limited conversions for target use, limited named exceptions for training conversion and for unfinished hulls to be converted to carriers)
There would have been cost pressures due to the end of the first world war and the upcoming arms race in warships that the treaty was negotiated to limit. Even if HMS dreadnought had survived till later, the naval treaty it would have required it to be scrapped or sunk. The dreadnought is actually named in section II of the treaty under British capital ships scrapped.
The 1930 and 1936 london naval treaties sought to extend the washington treaty.
Large naval trusts for preservation of major naval warships don't seem particularly prevalent in the early 20th century; battleships represented a large investment of steel/funds and were expensive to maintain even otherwise.
tldr; Older battleships were scrapped in accord with terms of the washington naval treaty which was negotiated to limit an arms race in warships