in the United Kingdom the Speaker remains strictly non-partisan, and renounces all affiliation with his or her former political party when taking office as well as when leaving the office. The Speaker does not take part in debate or vote.
Other than a coincidence of names and the nominal role of presiding officer, the two Speakers (US/UK) have little in common. The UK Speaker of the House of Commons is closer to the US Parliamentarian of the Senate, who advises the presiding officer of the Senate and makes certain decisions related to reconciling budgets with the House. In the US, the presiding officers are not non-partisan (the Senate rotates, frequently among rather junior members).
In the US, the Speaker is closer to being the Prime Minister, the one responsible for organizing the legislature. This is somewhat confused in that a Prime Minister in the UK also has some of the powers that reside in the presidency in the US. The US Speaker of the House is the head of the House of Representatives, as the Majority Leader controls the Senate and the President controls the executive branch. Tax bills must originate in the House, so Paul Ryan as Speaker of the House has more budget influence than Mitch McConnell as Majority Leader of the Senate.
Each cabinet secretary is responsible for developing a budget that the Director of Management and the Budget puts together into an overall budget proposal. Congress uses that as a framework (or counterpoint) for their budget resolutions. Congress has to pass (and the President sign or Congress override a veto) appropriations bills for each actual section. That is the legislation that actually allows spending. Without those, either they need a continuing resolution or the President has to authorize emergency spending.
Budget process references:
Another point of confusion is that in the UK, the Prime Minister is the consensus choice of the governing coalition. I.e. the other members of parliament choose the Prime Minister. In the US, the President is directly elected, so the President is not necessarily the choice of the Representatives or Senators even if of the same party. And of course, calling Donald Trump a Republican is itself questionable. He has more experience as a Democrat and gets many of his ideas from Democrats.
That's not to say that he has no points in common with Republicans. Just that he is by no means a conventional Republican, like Ryan or McConnell.
Trump promised no cuts to Medicaid
No. Trump promised not to touch Medi***care***. He never promised anything about Medicaid.