Some unis are far more focused on the PS as a form of selection than others. For LSE, most candidates (especially for Economics / other maths-heavy subjects) will have all A stars and do the PS is crucial. There will be info on the website and you really have to tailor it to their course. Be very specific! The PS is all LSE have to filter all A star candidates as no interviews and no admissions tests. The PS is a BIG deal for LSE. Loads of full A star candidates are left hanging until May every year. Some (like DD) never hear back at all which is ridiculous.
Cambridge have the SAQ so, if applying for say, Sociology at LSE, but HSPS at Cambridge, they can use the SAQ to talk about the Politics side of things and be much more specific about why they think the Cambridge course in particular suits their interests. Look at the modules and mention a few if possible.
Durham offer the opportunity to write a Durham-specific PS. This is because they have so many who also apply to Oxbridge, and they know Oxbridge don't want to hear about extra-curriculars / sports / hobbies etc on the PS (unless it's very relevant to the academic angle). But Durham (if I remember) recommend about 80/20 academic / EC for the PS. So basically they can rewrite the PS for Durham (submit this online once their application is acknowledged) and go on about how the ECs strengthen the application yadda yadda. They don't have to do this, but they can. Durham admissions are well known for being slow and cumbersome and come across as particularly disorganised, especially for large courses.
Bath have some quite unique courses so will allow an alternative PS in the same way - eg. if applying for Geography or Economics everywhere else, but International Development with Economics at Bath, they can write a second PS. Bath know their courses are a bit different and don't want people to be put off applying because of PS issues.
In our experience, Bristol we're very quick but sometimes what they do is put early applications 'on hold' - ie. they wait to compare with all other applications by the late Jan deadline.
UCL is a massive uni with so many courses and combos of courses and I think it's quite course-specific there as to who hears when.
That was our experience anyway from DD and friends.