There is no equivalent of your Primaries, I think that is a unique USA phenomenon.
As we don't have a president there is no point of contact between your system and ours in that respect. Other countries which have presidents (eg France & Germany) do not hold Primaries either.
The parties choose their ordinary candidates for parliament themselves. This is the same as your House of Representatives system, I think.
We have a "first-past-the-post" electoral system; which means that the leader of the political party which has gained the most seats in the Commons is invited by the Queen to form the next government. This person then becomes the Prime Minister and invites their senior party politicians to form the Cabinet; which is then the de-facto government.
Unlike how most American broadcasters seem to think, we Never use "Prime Minister" as a title (because it isn't one). The PM is Always called "Mr (or Mrs) xxx"; or, if you're really pushed, "The Prime Minister".
Proposals for Acts of Parliament (called Bills) are debated in the Commons by the elected representatives and then passed to the Upper House (the Lords) for debate and amendment by experts in the subject matter. It is commonplace for Bills to bounce beteween the two Houses several times before they're agreed.
When both Houses of Parliament have agreed and passed the Bill it is presented to the Queen. She is politically neutral and so has no axe to grind for any particular political party dogma. This tends to reduce some of the political excesses and polarisation that may otherwise happen.
Her Majesty scrutinises the Bill and can, at this stage, still suggest further amendments, but rarely does.
When the Queen and her Privy Council are satisfied that the Bill isn't going to change any more she will sign it. This is known as giving it the Royal Assent and from that moment it is no longer a Bill but it becomes an Act of Parliament and is entered on the Statute Book as the Law of the Land.
The European union plays a large roll in the decisions made in the UK as whatever decisions made about laws ect. within the European union must be abided by all the country's involved. The fact that the EU is an elected dictatorship argues against a true democracy as the people of the UK have non say on the decisions made within the union so it is not democratic.
So, the simple answer is: Yes, we are a democracy but in reality it does seem more like an Elected Dictatorship. Generally it seems to work but occasionally the actions of government do leave me wondering what planet they're on sometimes. It's not the same as yours and neither is perfect. We all have corrupt and incompetent politicians and we also have brilliant statesmen; but we can each get rid of the bad ones periodically so that's democracy at work.
However compared to other countries we have a wealth and freedom of speech and action that most of the nations of this world could only dream of.