Starting a project is usually good to write down what you do know.In this case, what would help any of us trying to assist you would be something that is as detailed as the information you already know.This includes names, dates, locations and any other tidbits as obscure as you might not think are important.
Take the following outline as an example and we'll see what you can fill in.
You (being online dialog, you probably don't want to provide any personal info about yourself, this would also go for any people below who are still living.This conversation can be taken private if necessary.)
Your Father, Name, birth, birthplace, death, deathplace, immigration dates, etc.Immigrations are important because they tell us when to stop searching one place and to start searching in another.
Your Mother, same info, including their marriage date, place.
Then we get to your grand-parents.Same exercise.Because we are looking for your James Beecroft, we should focus on him.You had the date 1915 for him, but I don't know what that date represents... so placing that on context would be a great help.
Was your mother an only child?Knowing siblings can often generate a link where the direct line goes cold.Do the same for your mother, and your grandfather if possible.
You mention Keighly, Yorkshire, England.My research has shown that Beecroft as a surname originates from this area.Within a short distance of Keighly, are many locations rich with Beecroft families and lines.Sorting these out can be a challenge.
Are there any other place-names that you've heard over the years that might be origins for your family line?These could provide the links you are missing today.
Genealogy as I like to describe it to people is like taking several large puzzles, dumping them all in a box, shaking it up, tossing a few handfuls of the pieces into the garbage, then attempting to put all the puzzles back together again by taking one piece out of the box at a time.Some days it will look like all you have are pieces and other days, magic happens and you find pieces that actually fit together.