Hi, my name is needtoknowneedtoknow!

Yes, claiming SDP isn't something that triggers a change to UC.

Firstly, are you sure you qualify for SDP?

https://www.gov.uk/disability-premiums/eligibility

Secondly, SDP can be backdated to the date you became eligible.

This isn't usually done automatically so you will have to request this. That request will go to a Decision Maker to decide when it gets backdated from. If refused, you can appeal.

You also need to be living alone (or you live with someone and they get a qualifying benefit, they’re registered blind, they’re a boarder or subtenant- but not a close relative, they make separate payments to the landlord).

Plus if someone gets Carers Allowance or Carers Element (UC) for looking after you, there is no eligibility to SDP.