Short Answer
Indirect evidence suggests that it is certainly possible that there were some veterans of Falkirk (1298) who fought at Bannockburn (1314). Men could be enlisted on both sides up to the age of 60; thus, for example, a 20-year-old at Falkirk would have been 36 at Bannockburn, well within the enlistment age limit.
Whether they numbered just a couple, or a dozen, or a hundred plus, on either side is impossible to say. It should also be noted that there were a number of factors (aside from death) which would have limited the number of possible participants in both battles.
Details
In Scotland, the servitum Scoticanum covered "able-bodied freemen aged between 16 and 60" so, whether Robert the Bruce used this or not to get soldiers, there is clear evidence that an able-bodied 60-year-old could serve. Robert's principal requirement was actually men who really wanted to fight ('wyn all or die with honour') so he would have welcomed any battle-hardened volunteers who had fought at Falkirk and who wanted revenge.
On the other hand, around 30% of the approximately 6,000 Scots at Falkirk were killed in the battle. Further, there had been numerous skirmishes and executions between 1298 and 1314, further depleting the likely number survivors from Falkirk. Nonetheless, the 16 year gap makes it plausible that some Scots fought in both battles. Even if we set an upper age limit of 40 at Bannockburn, men aged between 16 and 24 in 1298 could have fought in both battles. Robert himself, victor at Bannockburn, falls into this age group of 'possibles' (he was 24 in 1298), but the evidence indicates that he was not with Wallace at Falkirk.
In England, the range in the age of those who could be levied was also 16 to 60. The English army included four commanders who had fought at Falkirk (Aymer de Valence, Humphrey de Bohun, Robert de Clifford, Henry de Beaumont); they don't qualify according to the OP's criteria, but they brought enlisted men with them, some of whom would quite likely have been seasoned veterans.
However, somewhat reducing the possible number of English participants in both battles is that the English levies in 1314 were mostly from regions which had not been levied for Falkirk in 1298. The levies for Falkirk were from Lancashire, Chester and Wales but, for Bannockburn, Edward II had to cast a much wider net which included all the northern counties and the midlands; it is thus highly unlikely that any of the soldiers levied outside of Lancashire, Chester and Wales would have been at Falkirk.
On the age of soldiers, there is no archaeological evidence from Bannockburn to help us determine the age of the enlisted combatants. The best we can do is to look at the evidence from another medieval battle, Towton (1461), where
The remains of 38 individuals, including 28 complete skeletons, were
recovered. The bodies recovered from the Towton mass grave belonged to
men who ranged in ages from 17 to 50 years old; and had heights that
ranged from five feet to six feet, with the older men being the
tallest.
Source: 'A game of thrones written in bones: The skeletal collection from the Battle of Towton'. See also 'Osteological Analysis Towton Hall & Towton Battlefield Towton North Yorkshire'
The presence of older men at Towton suggests the possibility of there also being men at Bannockburn old enough to have fought at Falkirk.
