A number of Democratic members of Congress, Senator Bernie Sanders being the most prominent, support a Canadian-style single-payer system where doctors and hospitals are private, but health insurance is run by the government. But my question is, are there any members of Congress who support a British-style socialized healthcare system, where not just health insurance but healthcare providers like doctors and hospitals are part of the government?

Does Congresswoman Alexandria Ocasio-Cortez support such a system, for instance?

What the Government Provides Today

In the US as of January 2019, the government provides healthcare both directly and indirectly. Its direct programs (i.e. state ownership and employment of facilities and providers) are exclusively for members of the military, veterans or Native Americans, and are managed by the Department of Defense (through the services themselves), the Department of Veterans Affairs or the Department of Health and Human Services. Indirect care (i.e. health insurance) is provided through Medicare for the elderly; Medicaid for the poor (administered by the several states); Tricare for active duty or retired military; and a patchwork of other programs (e.g. CHIP, SCHIP).

Party Platforms & Politicians

As a party, the Democrats tend to favor either the existing ACA system or a "single-payer"/"Medicare-for-All" plan (which is typically loosely defined). The Republicans, for their part, tend to advocate for a more market-centered plan that relies on federalism and streamlining legal issues. Given the failure of the GOP in the last Congress to coalesce around any particular system, we can only say for sure that they oppose the ACA and single-payer.

Given that there are 435 members of the House and 100 senators, I would guess at least one advocates for or has advocated for a VA-for-all type of system. However, several web searches have turned up nothing. "Medicare-for-All" is a much more common phrase.

Ocasio-Cortez

Since the question asked about Ocasio-Cortez, during a recent interview with Jake Tapper on CNN, she stated:

Well, one of the things we need to realize when we look at something like Medicare-for-All, Medicare-for-all would save the American People a very large amount of money, and what we see as well is that these systems are not just pie-in-the-sky. Many of them are accomplished by every modern civilized democracy in the Western world. The United Kingdom has a form of single-payer health care, Canada, France, Germany. [Emphasis added]

What does she mean?

The UK, Canada, France and Germany each have quite different models for how they provide health care to their populations. The UK itself has a heavily-nationalized system (albeit with certain private providers) and Canada has a single universal insurance plan (i.e. true single-payer). Furthermore, France and Germany are both "universal multi-payer," with the state paying a large percentage of medical bills and citizens or supplemental insurance being responsible for the remainder. Given that, a fair interpretation is that Ocasio-Cortez is using "single-payer" as a shorthand for all "universal" systems, with the exception of the current universal American system under the ACA.

Summary

Without a concrete proposal, we can only speculate on what "single-payer" or "Medicare-for-All" actually mean, but a fair guess is something on the spectrum between France/Germany and Canada. While the UK is sometimes mentioned as having a "single-payer" system, it usually is mentioned alongside Canada and other Western countries, which have diverse health care systems. Few or no politicians are advocating "VA-for-All," which would be the closest American analogue to Britain's National Health Service, so we can fairly assume that direct nationalization only has marginal support in the US.