I'm going through the process of doing this right now. You can change the gender marker on your passport but that does not change your gender on your birth certificate. For that, you need a GRC. But it's very helpful anyway, because if (like me) you mostly use your passport to identify yourself, it'll show you as being male. Who ever sees your birth certificate? So it'll do for now.
I received my Deed Poll last week and have spent the past few days changing my details in the following order:
- Banks (so that I can use my old passport to identify myself as this needs to be done in person with a Bank Manager)
- Employer
- HMRC for tax & benefit purposes - you can do this online, it's really quick & easy!
- National Insurance Contributions Office (by letter, with a photocopy of my Deed Poll)
- DVLA for vehicle registration (fill in Section 6 of form V5C that you got when you registered your car)
- Mortgage company, car loan company, insurers, utilities, pension, cable TV, council tax etc. etc. (by letter, with a photocopy or Certified Copy as required)
- TV Licence (online) and mobile phone (in one of their shops by showing them the original document)
- Lower priority people, such as store cards.
- Once everyone is notified in writing (and I no longer need to identify myself by my birth name), my final step will be to update my driver's license and passport. I plan to do this next week.
My dentist let me change my details over the phone, and I simply filled in a form at my GP's office - they're even arranging me a new NHS Medical Card.
Some organisations asked for Certified Copies of my Deed Poll (such as some of my banks & insurers), but most were happy with an ordinary photocopy. So when I ordered my Deed Poll I ordered 5 Certified (or 'Legal') copies to send to various organisations, and then just made a bunch of photocopies (of the original) for everyone else.
I suggest you make a list of people and organisations with whom you usually do business, and ring up their customer service line to find what they need if you've changed your name by Deed Poll. That's what I did, then I created a spreadsheet so I could track all the details. Some will let you do it over the phone, others will require a letter.
My car insurance firm was a bit more complicated. Their underwriters needed to get their heads around the fact that my name and title are changing to male, but I'm still legally female for the purposes of the policy. It took a couple of days to get them to understand, but I had some great assistance from a CSR at my insurers and we sorted it all out (turns out one of his friends has transitioned so he was very sympathetic.)

Oh, and the only bad response I've had out of contacting the 32 organisations with whom I regularly do business? That would be my main bank's Indian call centre when I tried to update my online banking login details. The rep I spoke to simply refused to help me because I sound female but have changed my name to male. He completely stonewalled me and directed me to my local branch. Wouldn't even let me past security, despite the fact I answered all the questions correctly (naturally).
Hope this helps!