To change your name you don't really need a solicitor or anything like that.
All he will do is write on a bit of paper that you've changed your name to frederick jo bloggs and then he'll sign it.

All anyone needs to do in UK to change their name is to use it....

You are then using the name you legaly wish to be known by. Of course you must change all your documents as the new name will take effect in all aspects of your life, including banks, mortgage, NHS card, driving licence etc. etc.

The law as far as I understand requires you to inform banks... due to fraud...but not really needed in other aspects... typical... but seeing as the name change is on the passport and if you ever wanted to go for a new account or had to prove ID then you must use the name on passport, if they wanted further ID then other ID must match...obviously...

So in theory, you could have two names, but as long as you don't break the law in any way with them then it shouldn't be a problem...but it would be best to make any changes to all documentation.

You cannot change birth certificate... her passport can be changed back to it's original name if so desired but maybe it would be better to wait until you got the UK passport issued...I'm sure the passport office will realise and understand what's happened as long as you include a supporting letter.