There certainly are white forget me nots but the shape of the petals on yours are more narrow and elongated than the usual forget me nots , and the leaves are the wrong shape too. Couldn't find in my wild flower books.
Joining in to see what Eileen says...

I think it might be one of the mouse ears though the petals are usually notched.
They seed freely so left unchecked could become a problem.

going by the number of stamens [male parts 10 and styles 5 female parts] makes me go for common mouse ear Cerastium fontanum. Though usually the petals are bifid, notched.

Thank you Yorkslass and Seaburngirl for your quick replies. I have now read about mouse ear and it seems it is a food source for some moth species. However it says the seeds can last in the ground for up to 40 years and a single plant is capable of producing over 6000 seeds! I think I shall have to weed it out! That certainly sounds like it could become a problem, as you said, Seaburngirl. Thank you again.

it is indeed a food plant for a few moths and we have some of the close relatives growing for that reason.

Not convinced, but bow to the botanist...

No don't take my id for gospel, the petal lacking notches is what makes me doubtful but the no of reproductive parts is right and one of my books say notch size is variable.

Mouse eared chickweed I believe. I used to collect this and run of the mill chickweed for canaries.

Did you notice whether yours had notched petals Thorneyside?

However its a bit academic really as it shouldn't be growing in a flower bed unless you enjoy weeding...

I have pulled them out, found a third small one as well, so out they all came, before it poured with rain again!
I would be interested to know what near relatives you are growing Seaburngirl. I do try to grow as much as I can for the pollinators and insects. I dig up violets that appear and replant them where I want them and now have a little colony. Also have things like honesty, verbascum, evening primrose and red valarian, all grown from a few seeds collected on walks.