Billy
Seems a mystery & I do not have anything that can help shed any light.
I have seen a birth for an Edward Burley
Edward Burley Sep 1847 - St Luke London - Vol 2 Page 254
Edward Burley Mar 1849 - Lambeth - Vol 4 page 356
Edward Burley Mar 1845 - Chepstow, Gloustershire, Monmouth - Vol26 Page6
Didn't think these really matched (although my Burleighs are the Chepstow lot, there is possibly a link somewhere, but I am still to find a Burleigh)
Then found the Edward you refer to on the 1861 census, birth Cheltenham, Glous, Mother Sarah, brothers John E & William H. Spelt BURLEIGH. All living in Paddington. Edward aged 15, his occupation a railway porter.
Think I have then found this same Edward on the 1871, this time with a sister Christina, who must have been away on the 1861. No William on this one, but John is & Sarah with 2 visitors. Edward is an unemployed Porter.
Not sure if this is really any good or will just confuse matters.