But if he does have property or assets in his name only, and he doesn't write a Will, 1/3 of his estate automatically goes to his children upon his death. He can have all the verbal "understandings" that he wants with his wife, that won't change the law, or what his kids are legally entitled to upon his demise.
If the OP is an only child, or 110% certain that they and their siblings are 110% ok with waiving their inheiritance in favour of their mother, everything should work out ok. But all it takes is for one disgruntled sibling to feel that they are being done of of what is "theirs" for a nasty situation to arise. Your dad writing a Will leaving every thing to your mom eliminates that possibility.
Writing a Will is not expensive or risky, providing you use a reputable solicitor. The solicitors fees will be a few hundred euros, if even that. If your dad is a member of a Credit Union, they often offer this service for free. There are no taxes or other costs involved just for writing a Will. Inheritance taxes kick in once your dads assets are being dispersed to his family after he dies. These will be the same whether he writes a Will or not.
If your father decides to have the solicitor who wrote the Will be the Executor of his estate upon his death, he can. Their fee is usually 1-2% of the value of the estate. Or your dad can appoint a capable family member or friend to be the Executor, and save the estate some money. The Executor can be a beneficiary of your dad. It can be you if he wants. It is not a hard job, especially if your dad passes away with no major debts or charges on his estate, and the estate comprises fairly normal things such as the family home, bank accounts etc etc. The folks in the Probate Office are very, very helpful when it comes to helping inexperienced Executors navigate thru the task before them.
Best of luck to you at this difficult time.