Is it true that wales is essentially the ethnic remnants of the Roman Empire and the rest of England was ethnically Germanic tribes who crossed over the English Channel?
No, that's not true. I will go through a bit of a history of ethnicity in Southern Great Britain but keep in mind that while the cultural and ethnic identification of the people of the British Isles over the past 2500 years is relatively knowable, the genetic makeup is not as well known in all cases.
Celtic Britain
2500 years ago England and Wales were largely populated by Celtic Britons who spoke Brythonic languages.
Roman Britain
About 2000 years ago the Romans conquered England and Wales. There were Roman settlers, but these settlers mostly would have been in a handful of urban areas in England. During the 400 years of Roman Britain, a distinct Romano-British culture developed, but it was more prevalent in Southern and Eastern England than it was in Wales. During this time period there would have been three populations, with a continuum of Romanization running through, particularly the latter two:
Settlers from other parts of the Empire and their descendants. This would have been a tiny minority.
Culturally Romano-British. These would have been a significant population, probably the majority in most of the cities, and maybe a majority in some or all rural parts of Southern and Southeastern England.
Culturally Celtic. These would have been the majority populations in Wales, Cornwall, and Northern England.
Anglo Saxon Britain
Approx 1500 years ago Germanic tribes started arriving in significant numbers, these would be the Angles, Saxons, and Jutes. They would gain political control over most of England (but not Wales or Cornwall), and the earlier Romano-British or Celtic inhabitants of England would assimilate into the dominant Anglo Saxon society, but the Anglo Saxon settlers would not ever demographically be the majority in England (although maybe they would be in areas like East Anglia).
Danelaw
About 1200 years ago Vikings started raiding and then invading the British Isles. In England it was mostly Danish Vikings. While these Vikings had significant political control in some areas for 200+ years, I don't think they left much of a demographic footprint.
Normans
Approx 950 years ago the Normans (Settled French former Vikings) invaded Britain (both England and Wales), they imported a Norman elite but that did not comprise a significant proportion of the population. The Normans that did arrive mostly ended up in England, although an exception to that is Pembrokeshire in Wales (AKA "Little England beyond Wales").
Summary
Wales probably is most demographically similar to pre-Roman British populations. The majority of English ancestry probably comes from these populations too, but English would likely have more Roman (which could mean people from basically wherever in the Empire), Germanic, Scandinavian, and Norman mixture.