Do I need to add a number prefix when I am not in the uk

For all phone numbers, whether for local or for international consumption, you can always use the international version, with the + and country code.
Without the international code, they can not see which country the phone number is for and might assume it is for a different one, which will result in a failure to reach you.
You do NOT need to change the number, your phone number does not change while you are abroad, it will keep the +44 start.

If there is a country with a phone system where the + does not work, as it does on land lines, they know what to replace the + with.

The best way to add your phone number and address to your bag is to do it in a closed tag, which can be opened when needed but which does not show your address and phone number to all.
Besides, you can use a 'home' address which is not yours but someone who will be home while you are traveling.

There are special tags around which will hide your address, but you can also use a normal luggage tag where people can see what you write.
Just write your name and mobile number (or just name and 'information inside') where it will be visible and more information on the back.
That can be your holiday address, a home address where someone will be while you are on holiday, your work address if work is alright with your case being dropped off there at need, or even your home address if you have no alternatives.
It is also easy to make your own if you can get hold on suitable card or plastic, write, fold over and make a note on the outside it holds your details. You can tape it closed or punch a few holes in and thread a cord though.

Your (mobile) phone is a UK one. Thus, to unambiguously identify it as a UK phone outside of the UK, you need to add the UK’s country code — whether you are in Dubai, Ireland, the US or Australia.

Never ever leave out your home country’s country code from your phone number if you are going abroad. Many countries prefix their area codes with a 0; in Germany, the area code 0770 corresponds to the area South of the towns of Villingen-Schwenningen and Donaueschingen. Thus, always write +44 for UK.

If you added the country code for Dubai and somebody attempts to call you, they would end up calling a Dubai phone.

If you want a person located abroad to call a UK number, then yes you need the +44 prefix. There is no way for a foreign telephone company to know magically that 077... is a UK number.

Or do I need to put the country code of Dubai?

Your telephone number will stay the same wherever you are in the world, no matter which network you connect to. (Unless you change the SIM card of course.) You don't adopt the local prefix.

so that airport staff can contact me if my luggage gets lost.

Airport staff are not going to look at your bag or call the number on it if it gets lost or misplaced. The baggage handlers use the barcode they print to direct your baggage around, and if those get ripped off, they just put the bag away until someone comes looking for it.

As pointed out by Willeke, taxis and hotels may indeed take the time to call the number, or check that the name on the bag is yours when you pick it up.

Also, I know you did not say this, but do not write your address on your suitcase. That just tells people that your address is likely to be unguarded for the next week or two.