(Update: new summary)
I have given in to my weaker loquacious side and allowed this answer to become a tar baby, incorporating many topics that are only tangential to the OP. The summary is that bicameralism is one of many governmental architectures designed to incorporate stakeholders, foster deliberation and slow consensus. Although it was a tool familiar to the founders, they considered a wide variety of models of government, including explicitly the Iroquois, the Polish, the Roman and the English.
There are a whole bunch of assumptions in this question that don't pan out. First of all, France had a parliament, although it served a very different function. The Swiss and the Dutch were both republics, although substantially different from either the British or Republican model. Russia had the Duma. Poland had a bicameral parliament in 1493. I don't remember enough of the structure of the Spanish or Portugese government to make any statement.
(The following paragraph has been rewritten at the request of OP and others)
I also question your assumption that the Founding Fathers would have leapt to emulate the British model. Remember that their first attempt (the Articles of Confederation) was very different; it was unicameral, very limited in scope and effectively included no executive. The discussions about the second attempt (the Constitution) included several different solutions. The Virginia Plan was bicameral, but the upper chamber may not have been directly elected (wikipdedia disagrees, and I don't have my other sources with me). The New Jersey Plan advocated a unicameral legislature. Hamilton advocated a government that was explicitly constructed on the British model - his plan was soundly rejected, partly because it called for a strong executive serving for life, and partly because the delegates feared Hamilton's affection for the British Model.
Bicameralism was a constraint of trying to include the right class of stakeholders, not a slavish adoption of a British model. The fundamental problem of the Constitutional Convention was to find a way to build a government that was stronger than the dysfunctional Articles of Confederation, but not so strong that it would subsume the state governments. This is not the problem that the British Parliament was designed to solve.
The British Government wasn't "fixed" at this point; it was still being redesigned. Yes there were two houses of parliament, but the relationship between those houses, and the relationship of the houses with the Executive/monarchy were still in flux. Their electoral process doesn't match anything we'd recognized (rotten and pocket borough's for example. I can't cite, but I believe Jack Rakove lectures on construction of the government which available through iTunes, and are both full of knowledge and a pleasant listen, touch on this fairly well; they also include a reference to one of the best books on the formation of the constitution, the name of which escapes me right now. The British Parliament is transformed by the American Revolution (the Intolerable Acts are important from the British point of view), and that transformation continues throughout the French Revolution and the Regency Crisis.
(Update: I mistakenly included the Albany plan in my original writeup, because I'd confused it with the Virginia plan; the Albany Plan was a 1754 plan for a defensive alliance between the colonies. Based solely on the Wikipedia page, it appears to have been unicameral. The relationship between the Albany plan and the Iroquois is significantly more complex. I need to read Forgotten Founders in more detail. Another Yahoo asserts that the Articles of Confederation are a version of the Albany Plan, but
I'm afraid that I can't follow the author's line of reasoning until I've done a bit more research. It appears that the League of Five Nations, the Albany Plan and the Articles of Confederation are all solutions to how to compromise sovreignity with confederation - to permit member states to ally and share governance on common matters, but to retain agency within their borders. That's a fascinating and relevant discussion, but is not precisely germane to the question).
Jefferson and his Democratic Republicans (although that term did not yet exist when the constitution was designed) despised the British, and would have resisted including anything British.