You should first update those records that are most important to you. Most people update their passport first because they view this as being a quasi-identity document. This may be a good start, as HM Passport Office are renowned for carefully checking that you have all necessary documents in place before they issue a passport. As a result, some other organisations often accept a new passport as strong evidence of your change of name, and in some instances will not insist on seeing your passport or Deed Poll, if you give permission that they check directly with HM Passport Office.
However, in some instances, HM Passport Office (and/or the Foreign Office) will insist that you send them some other evidence of your name change before they will update your passport. This would be the case if they did not believe that you really intended to change your name for all purposes. This might, for instance, be the case if your new name is very unusual, or if you are changing the gender in your passport.