A friend of mine who did her PhD has told me that she picked her PhD project title from FindPhD and was not asked to submit any research proposal. However, whenever I find an interesting project, it says that I must submit a proposal and I have no time to write one. Can anyone explain this to me? Thanks.

I think simply that it varies by institution and probably whether funding has been sourced.

I just had to apply through UK Pass for mine and if I remember had to write a cover letter, personal statement and include references and academic transcripts. Because it was studentship, the research proposal and funding had already been done for me.

Depends on the funding method.

In the UK a lot of supervisors write a project title that fits their research and they get funding organized themselves for the project before advertising it. Then they are looking for a student to do that project where they are looking more at the student (Though I did this and was still asked to write a mini-proposal as part of the application). You have a bit less academic freedom but it means the money is there and usually, you have a good research question with some sort of plan.

Or you can write your own proposal and apply for funding yourself. But I don't know much about that method except it can be competitive.

Presumably because the deadline is imminent...

You may have to quickly hash something together and get as much feedback as you can from others on it (especially helpful since you don't have a lot of time) or else wait til the next round of deadlines... I'd go for it - you just never know!