Many enthusiastic responses, but no-one yet has tried to answer the question.
Firstly I note that the oldest the tree can be is 2014 -1977 = 37, plus the age it was when the TPO was made. The age it was then must have been low enough for it not to have been included in the Order. Although the Conservation Area rule of 75mm diameter doesn't apply to area TPOs, you could argue that any tree that small couldn't have entered the COuncil's thinking when making the TPO.
OK, so the Forestry Commission a few years ago published data for ageing trees. Basically, until 'mature state' trees are in 'core development' and put on the same amount (subject to weather fluctuations) of diameter every year. For Beech the lowest 'first mature state' ring is at 60 years.
The next thing is, the amount of diameter depends on situation. The annual increase in diameter in milllimetres is -
Champion tree potential (ideal site conditions) 12
Good site, open grown, sheltered 10
Average site, garden, parkland 8
Churchyard 8
Poor ground and/or some exposure 8
Inside woodland 6
So you could decide which of these categories represents the conditions of your tree for most of its life. Then measure the diameter of your tree and divide by the relevant number. That should be the age. Then take the category number again and divide it into 75mm. That will be the age that should be deducted for the early years cut-off assumption.
Here's an example (I'm confusing myself so this will help me too). You have a Beech on an average site, it measures 400mm in diameter. The data suggests it will have put on 8mm diameter a year. 400 divided by 8 is 50 years. But it was not TPOable until it was 75mm/8mm = about 9 years old. So its TPOable age is 50 - 9 = 41 years.
In your case the TPOable cut-off would have been 2014 - 1977 = 37 years. So the tree in this example would have been just big enough in 1977 to be TPOable (it would have been 50 - 37 = 13 years old and would have been 13 x 8mm = 104mm in diameter).
Further advice - check that Copper Beech and Common Beech have the same growth rate, or else the FC figures are useless to you. Also, assume self-seeding rather than planting. Check that site conditions haven't changed markedly since 1977, including whether the surrounding trees were big enough back then to suppress the Beech. Don't use these figures if the age seems to exceed 60 years. And most importantly, don't blame me if it doesn't work out, I can only advise on generalities and am quoting the Forestry Commission.
As ever [sigh] I have had to assume you are in England/Wales. If you're not, the rules are different.