Carol
By law, births are registered in the district where they happened.
It would be highly irregular for a nursing home to be involved in maintaining incorrect or fraudelent birth records or arranging sinister/underhand adoptions.
Have you had a look at the birth indexes?
www.freebmd.org.uk (but not complete or
www.familysearch.org if you don't have an Ancestry or FindMyPast subscription)
If you search for Fairweather/Fairweather births in Lincolnshire after 1950 there's yours but also a boy born the year after. Your birth is in Lincoln district, so is the other.
There are a few earlier and later births but in Boston district.
So your birth combination of names is quite unusual although there are other Fairweather births but with different mothers maiden names.
If the stillbirth enquiry comes back negative, then it might be worthwhile considering the 1963 birth.
Anyone doing your family tree for legal reasons would be remiss not to buy the certificate even if it was for elimination purposes.