It was very rare to find any royal family that could be classified as truly native, given their habit of intermarrying. For example, the House of Hanover are often characterised as "German" because they ruled the Electorate of Hanover, but George I's ancestors included Danish and French monarchs, and the "Scottish" Stuarts had important ancestors from France and Germany. So the premise of your question is quite shakey, because the obvious question is, what does it mean for a monarch to be considered "English" (or Scottish or French or German)?

The Tudors were Welsh. Henry VII, the first Tudor king, was born in Caernarfon to a prominent Welsh family. Throughout his life he described himself as Welsh including to his biographer.

I'm not entirely clear as to how you're using the ethnonym "English." I mean, the house of Windsor has been ruling England for 100 years longer than there has been a USA, so surely they count as "English?" If you're seeking a dynasty which is native to the territory from its inception, why are you looking at the English, that is, the Angles, rather than the Britons of the Roman era? Not that they were the very first population group there anyway.

In any case, you need to define your terms better if you want a good answer.

Depends on how you want to define them and their members and whether you want to stick strictly to their house origins or the actual members of the house and their lives.

• Saxe-Coburg und Gotha/Windsor: Despite their initially German name and descent from a German (Prince Albert) this house is pretty damn English, all born in England and raised as English/British. Since 1917 they divested themselves of their German possessions and titles and recast themselves as wholly British. Despite this they do continue some German traditions such as giving Christmas presents on Christmas Eve instead of Christmas day.
• Hanover: A German house by origin and undeniably pretty German during their early history in Britain but by the time of George III they had pretty much gone native. George III was born in England, spoke English as his first language and never visited Hannover. Once Victoria ascended to the throne and they lost Hannover, they were mostly English/British even if they still retained German titles and often married Germans (see Prince Albert).
• Stuarts: Scottish although like the Hannoverians went native although it must be said they did have some distinctly non English attitudes towards the role of the monarchy and amongst the later kings, religion.
• Tudors: Welsh although were going native from as early as Henry Tudor's marriage of Elizabeth of York. There wasn't much Welsh about Henry VIII onwards and technically speaking Wales was annexed into England during Henry VIII's reign.
• York/Lancaster: Cadet branches of Plantaganet although they were very English by this point.
• Plantaganet: French House but which became increasingly English, especially once the French territories were lost. By the later kings they were only a French house by name, not by culture, land holdings or attitude.
• Anjou: French
• Blois: French
• Normandy: Norman French
• Wessex: Anglo-Saxon / English

So it all depends on how you want to define it. The Windsors are undeniably English in 2014 but if you want to be pedantic they are a German descended house of royalty, regardless of their 1917 rebranding as 'Windsor'. Going further back the House of Hanover became very English by the time of George III and even more so once Hannover was lost but again they're technically a German house. The Stuarts were Scottish, the Tudors originally Welsh and both tended to marry abroad but again once they came to power most were born in England, grew up in England and were educated in England.

So if you wanted to find arguably the last 'English' monarchs in England then you have to go back to York/Lancaster in the late Middle Ages and then before them the house of Wessex during the late Anglo-Saxon period but this is entirely ignoring the people reigning. As far as I'm concerned the monarchy has been a natively English institution since George III and was English before the Stuarts came to the throne, regardless of the Tudors Welsh origins. The later Plantaganets were also English since arguably Edward III becoming the first King to address Parliament in English but you could go further back than that if you want to date it back to the English Kings losing their French possessions.