There will be no inheritance tax implication if your Fathers estate does not exceed his NRB which is currently £312k for a single person, but as he can also claim your Mothers unused NRB this effectively doubles his NRB meaning he could leave up to £624k without being liable for inheritance tax.
If he gifts you £100k there is no tax implication on this outright gift, even if he died before 7 years as he has not exceeded the above allowance.
If he buys a share of your flat and lives there as his main address then there is no Capital gain tax to worry about........ but if he buys a share and never lives in the flat, his share would be subject to CGT when the flat is sold.
It is better that he gifts you £100k, as it will not complicate his share of the flat. i.e. if he dies he would have to leave his share of the flat to you in his will, or to have bought the flat with you as "joint tenants"......rather than "tenants in common".