hi!
so i applied to oxford, bristol, UCL, manchester and sheffield last year (sent application on 24 sept) and wondered the same thing.
for oxbridge, they usually need your LNAT to be sat before the end of october (mine was before 24 october 2019 for oxford) because the LNAT scores are sent to your universities in two batches, essentially. I'm assuming its the same since Cambridge is one of your options which is an earlier application than the others so you will need to ask them for the exact date by which you need to take the LNAT if they havent given you one. Cambridge will need your results in the first batch they are sent out so they are able to consider your application in full and decide if they want to give you an interview for december. i found this on the LNAT website for this year:
"Starting on 21 October, the first batch of LNAT scores (tests taken between 1 September to 20 October) are released to the universities to download. After that date, within 24 hours of finishing your LNAT, Pearson VUE will make your LNAT score and essay available for download by your chosen university or universities. They will see your score before you do."
I appreciate that the exact dates may vary for you, so have a look on their website or contact cambridge university about it. they will not consider your application if you do not take it in time.
as for your other universities, i dont think they have a date by which you need to take the LNAT and so they will just consider your application as soon as theyre able to see your score and essay. if you get it done asap due to cambridge, this may be quicker too.
best of luck, and make sure you book it as soon as you can before it fills up! i had a friend who took his test too late and oxford didn't consider his application as a result.