Apologies if this has already been done, because I've only skim-read the thread, but it isn't as simple as just 'adding your name to the deeds'
I had the same situation with Ex-DH and we got legal advice on it, and this is broadly what we were told.
If there mortgage is only in one name, the title deeds can also only be in that name. The mortgage company will not allow another name on the deeds.
You therefore have a couple of options. You can register a Deed of Trust, which gives you a legal standing regarding the house. Any solicitor can do this, and you can (I think) either register a % split, or or have a joint tenants/tenants in common distinction in the document.
Option two is to remortgage, take out a joint mortgage and get you added onto the deeds.
But if you do this, you are liable to pay the stamp duty on the entire value of the property (not just the 50% share that you are now 'getting'). So if your house is over the stamp duty threshold, this could be pretty pricey!
Also, if you live in a leasehold property, you'll have to get the lease amended, which is another expense, especially if your freeholder likes to fleece you for every little possible charge possible.
Obviously it is worth getting proper legal advice on the process for either of these options, but it was more complicated than we expected it to be!