As for the U.S., my impression is that most of the legislation that is passed at the federal level begins as a bipartisan effort.
I have never heard such a thing reported in the UK.
I also suspect that individual members tend to work well together, irrespective of party, when the cameras are off. For example, if a minority member of a committee has a proposal for a bill, he or she would probably hand it over to a member of the majority party because the majority member can move it more easily. Then they'll take joint credit.
In the Uk there is no 'author credit' to a bill. It is either a government bill or an opposition bill. parliamentary time is allocated by the government, which gives most to its own bills, and a little to the opposition to allocate for their own purposes. A tiny amount of time is available for backbenchers of any party to propose something themselves and there is an annual ballot. A very few short laws may pass this way, provided there is virtually no opposition from anyone. Again it sounds to me as if the US is much more person centred and less party centred.
The commons has a number for committees which investigate things, including to an extent how the government is doing. these are stacked with a government majority and tend to agree with the government, but not always. However, they are unlikely to accomplish much in terms of creating or originating laws. There are civil service committees which plod along making suggestions for reform of laws in the background.
A significant volume of time is occupied rubber stamping or customising legislation which has arisen at the EU level. In this respect, I suspect MPs are conscious of an endless stream of regulations they have to vote on and pass, which are occupying their time, which come from the EU. maybe they wanted to leave the EU because they resented this. As a process it hardly ever comes to public attention.
On the big ticket items, however, like "Obamacare," there's nearly universal party discipline.
Very few matters in the Uk are not 'whipped' with the parties expressing an official view for everyone to follow. The commons has around 100 /650 members not from the two main parties but mainly these are regional parties, the biggest chunk being Scottish nationalists. UK government is essentially 'winner takes all', majority of 1 in the commons decides anything. A government which cannot maintain a constant majority normall falls, so governments seldom bother negotiating with the opposition which will automatically vote against them.
Where in the rumble remaining from the nuclear counter-strike, do you envision the congressional proceedings taking place?
Can't speak for the US, but no one in the Uk believes a Trump president would actually cause a nuclear war. If anyone here is upset about nuclear war, it is the right wing conservatives objecting that the current leader of the Labour party might refuse to permit a nuclear strike on someone were he to become PM. Techically, control of the army rests under the monarch as do many powers. However all of these are exercised in practice by ministers appointed by the monarch following the instructions of the prime minister. The monarch can technically name anyone she pleases as prime minister, but has chosen not to do so for a very long time. We do have a history of disposing of uncooperative monarchs.