Some more information required........
1. Is there a mortgage on the property?
2. If so, what is the capital figure outstanding?
3. Do either of you live in the property?
Assuming- rightly or wrongly- that you live in the apartment and there is no mortgage, your father can transfer his share to you.
You will not have a tax liability. Your father will, potentially, have a CGT liability if the market value of the property has increased since purchase. It would be on the value of his share only.
And , yes, you both need separate solicitors.
mf