Are you actually inheriting a house - i.e your name will be put in the deeds as the new owner - or are you inheriting the proceeds of the house as part of the estate?
I have no idea of the answer to this question, but if you are inheriting the money from the sale of the house, handled by the executor, who will give you the money, not the house, does that not make you a first time buyer?
In which case you won't pay stamp duty on the first £300.000 of purchase price, even after the June deadline. FTBs are exempt.
If you are not counted as a FTB, beyond the June deadline there is a halfway measure - you don't pay stamp duty on the first £250,000 of the price.
So depending on whether you are a FTB, and on the price of the house you would like to buy, you may not pay any or much stamp duty anyway.
I think you could really struggle to complete by the end of June. You might be lucky, if you find a property that is chain free and empty, if the searches come back quickly (they are taking ages) , that you can find a solicitor ready to help you try (many have cancelled holiday to meet the current frenzy and will have lots of staff on hol after easter), if the inherited house sells quickly and probate happens quickly, OR if you go the mortgage route, you find a house for which the lender finds no quibbles, case for retention or works to be done, etc, and if their solicitor moves quickly.
As a cash buyer you will be in a far stronger position to make offers. Vendors like a cash buyer - not lender to impose conditions and be slow.
I would avoid all the costs associated with getting a mortgage, the stress of trying to beat the stamp duty deadline, and wait until you can proceed as a cash buyer.