Since the Oxford application deadline is on October 15th, does that mean that I also have to submit my other applications to other universities on that date (October 15th)?
No. The Oxbridge deadline is October 15. All other UK unis are in January as far as I’m aware. You have plenty of time. You don’t have to apply all at once. You could apply to oxford and then wait a bit to apply to other places if you wanted
Since the Oxford application deadline is on October 15th, does that mean that I also have to submit my other applications to other universities on that date (October 15th)?
No and it is sensible not to do so.
Oxford decisions are being sent out on 14th January (obviously if you don't get an interview you will find out sooner)
The last day for equal consideration for other non-medical applications is 6PM on 15th January,
If you don't get an Oxford offer you are likely to want to apply for other high tariff courses elsewhere
If you do get an Oxford offer (which you want to accept) you are really looking for an insurance offer. There is no point in applying to courses with tariffs equal to or higher than Oxford which have no reputation for leeway for near misses.