If you are considering changing your solicitor, it's a good idea to do some research and find out which law firms specialise in asbestos cases. Even better, find out which firms have been recommended by people who have been through the same thing.
Changing solicitor should not cost you anything. Speak to your potential new solicitor for reassurance on “no costs to you on transfer”.
You do not have to contact your current solicitor to tell them of your intention; your new solicitor will give you a form of authority to sign, which s/he will then send on to your old solicitor.
Your new solicitor will ensure that your old solicitor gets paid for the work s/he has done so far on your case, but only when your case has been concluded. This is known as a lien on costs.