Mandatory Palestine was not a British colony, but a mandate territory that was governed for the League of Nations by the United Kingdom. As it was not a part of the Crown's dominions but merely governed on behalf of the League of Nations by the United Kingdom, it was from the viewpoint of nationality law, foreign soil and birth in Palestine did not make a person a British subject. It had
its own distinct citizenship (i.e non-British citizenship) between 1925 and 1948.
To the best of my knowledge, Crown Service of itself has never conferred British citizenship/CUKC by itself. Your father's Palestinian passport would have been that of a
British Protected Person*, not of a British subject/CUKC. If he acquired any other citizenship (either Israeli or Jordanian or any other), he would automatically have lost the BPP status.
* - Pre-1948 Palestinian citizens were prerogative BPPs, not statutory BPPs.
Even assuming that the BPP status is retained, it does not give any right to reside in the UK. It is subject to immigration Control and you will still need a visa to enter the UK.
So, to look at whether you may have acquired British citizenship, you will have to see if your father was registered as a British subject/CUKC before your birth. Mere proof of Crown Service is not, in my opinion, sufficient.
Further Reading:
"Mandated Territories" and "Trust Territories"
Protectorates and Protected States
Chapter 54: British protected persons, general information (nationality instructions)
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.