Hi Mike,
not certain as needs further work, but on the face of it and to my great surprise, yes, it does indeed seem possible.
A very quick look at the British History website states;
http://www.british-history.ac.uk/vch/glos/vol5/pp300-325#h3-0004

Lydbrook;
"The church of Holy Jesus, which stands high up on the north-east side of the main valley, was opened in 1851. New building at Upper Lydbrook in the early 20th century included a chapel and a school."

Bilson Green aka Lower Cinderford
"Elsewhere in Cinderford building continued in a haphazard fashion after 1840 and was mainly confined to scattered encroachments made before 1834. At Bilson industrial development included gasworks erected in 1860, and a large school was built north of the road to the town centre (later Station Street) in 1877."

(This makes me feel very old, as I attended Bilson Infants & Primary school c1970).

More to follow later.

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UPDATE:
First thought, sorry, hadn't fully realised we're talking about 7 year old infants here, so they may not be able to travel miles for school each day !

Secondly, the above link to the "Education" section of the British History site, in an almost "typical" manner for that site, appears to then contradict that site's own Summary I quoted earlier. This IS an excellent site, it does have lots of great information, but it does require carefull reading all thro'. Sometimes I think it's been edited with new information being added haphazardly and without them checking & editing the earlier sections to suit the new text.

Please read the whole "Education" section Mike, but I now see

"Following the Education Act of 1870 there was an acute shortage of school places within the Forest. The voluntary system supported ten church schools and there was an unknown number of private schools. In 1874 the Education Department, in giving notice that a school board would be formed for the Forest if enough places were not supplied, identified ten districts in want of schools or increased accommodation." Annoyingly it doesn't then clearly list those districts... but later says;

"The earliest day schools in Cinderford were evidently small private schools such as the seven recorded in that part of the Forest in 1833. They taught between 7 and 30 children and five had been started after 1822.....

Following the opening of a board school at Bilson in 1877 attendances at Cinderford's older schools declined. Flaxley Meend school, which was kept open principally by a local surgeon William Heane from 1878, became an infants' school and closed in 1881. A school board formed for Flaxley Meend later that year lapsed in 1883 when the area became part of East Dean township.

Bilson school, one of the first to be provided by the Forest board, had new buildings in the later Station Street with 433 places in boys', girls', and infants' departments and a detached pair of schoolhouses. The school was enlarged in 1879 and 1886 and the average attendance was 670 in 1889 and, after the older children had been moved to Double View school in 1896, 542 in 1910. Remodelled in 1914, it had junior mixed and infants' departments from 1932 and an average attendance of 429 in 1938. From 1974 it took only infants, the juniors being transferred to new buildings in Latimer Road, and in 1992, as Bilson County Infants' school, it had 190 pupils and Latimer school 251."

----

"In Lydbrook, where a schoolmaster was living in 1837, a day school was established in the C. of E. schoolroom at Upper Lydbrook in 1849. At the end of 1851 the building housed separate boys' and girls' schools or departments with a combined average attendance of 92. They were funded by an annual grant from the Crown, voluntary contributions, and pence and were managed by the minister of Holy Jesus church, which had been built next to them. In 1873 new rooms were added for the girls to leave the original building for the boys, and by 1875 an infants' class had been started. At that time there were many dame schools in the area, including Worrall Hill, and a few years later a board school was provided at Joy's Green to meet a continuing shortage of places. The church school, which was reorganized with junior mixed and infants' departments, had average attendances of 193 in 1889 and 185 in 1904, by which time the county council had decided to replace it. Disputes over the choice of site for the new school delayed its construction, which became more necessary with overcrowding at the church school, and in 1908 the older boys were transferred to a temporary school established in the Baptist schoolroom at Lower Lydbrook. The council school opened at Upper Lydbrook in 1909 in brick buildings with places for 246 children. Average attendance was 210 in 1910 and, after enlargement, 251 in 1938, just before the junior and infants' departments were amalgamated. As Lydbrook County Primary school it had 108 children on its roll in 1992. The church school was abandoned in 1909 and its building reverted to use as a church hall and Sunday school. It fell into disrepair and was pulled down in 1975.

Joy's Green board school, built for the Lydbrook district, opened in 1883 in new buildings including a detached schoolhouse. The school had junior mixed and infants' departments with places for 227 children and in 1889 the average attendance was 200. The building was enlarged in 1894 and the average attendance was 287 in 1910, falling to 110 in 1938. As Joy's Green County Primary school it had 58 children on the roll in 1992."

excerpts from http://www.british-history.ac.uk/vch/glos/vol5/pp405-413

So, there seems little likelihood that infants from Lydbrook and Bilson Green Cinderford would attend the same school. There were other schools between Lydbrook and Cinderford, such as at Steam Mills, so... ? I suppose it's possible this happened if their parents were able to fit it into their daily lives, so would perhaps need to be fairly well-off and probably quite influential ?. However in this case that would be the gas works manager's son, and they lived very near Bilson School...

??

Do you have any reason for thinking they did go to school together, Mike ?.