I am an Englishman who avoided fighting in France during the Great War. What social stigma might I face?

To a degree it depends on where you lived, but largely things weren't great for those who resisted. Everyone had a friend member or a friend, or at least knew someone who had a family member or a friend, out fighting at one of the Fronts, so there was a lot of social pressure to support the war. If you were a healthy male of military age who did not want to go you were seen as taking advantage of the sacrifice of men who went. People also perceived them as cowardly, and feminine.

The government, the press, and supporters of the war emphasised the sacrifice made by soldiers, and their heroism, and created an image of manliness linked with soldiering and sacrifice for ones country. If you did not conform to that you might be shunned by the people you knew, abused on the street, and even beaten. Something like 70 men who were conscientious objectors died through various injuries, mistreatments, and suicides during the war.

During the first couple of years of the war the British relied on the voluntary system. Basically men had to choose to join the army. However, due to massive pressures from sections of society, government, and the press, conscription was brought in by the Military Service Act in January 1916. If you were a male of military age who did not want to fight you had to publicly voice your stance at a Military Tribunal and get judged as to the worthiness of your claims by notable locals and a military representative. People wanted to avoid service for a variety of reason. The most common were having a business that could not run without you, or a dependant family member. Although, there were people who just did not want to fight in a war because of their beliefs. There was an poorly defined and thought out clause in the Act that allowed men to avoid military service on grounds of their conscience. Basically it meant that the tribunals had to decide whether or not you held genuine beliefs that meant you could not kill, and then whether or not you were suitable for exemption from duty, non-combatant service, or actual combat. These tribunals were incredibly biased, essentially they wanted to send as many people as possible, and usually could not abide those claiming exemption of grounds of conscience. Thousands of men ended up in prison, or were sent to the military and subject to military discipline.

Those who this happened to often suffered horrendous abuse. Men in prisons at this time were often poorly fed, and depending on whether or not your guard was a sadist you might get frequently beaten. Those who were sent to the army but still refused to obey orders received similar treatment. They would be in and out of jail, and occasionally tortured. One man, I think his name was John Grey, was famously beaten until he said that he would be a soldier. They stripped him, kicked the living hell out of him, and repeatedly 'ducked' him in a nearby body of water. 50 or so COs were actually sent to France, subjected to military discipline, and then sentenced to death for not following orders. However, they were pardoned on the day the were supposed to be executed.

These abuses were published and became common knowledge, and people started to see COs differently and appreciating that they weren't necessarily cowards if they were willing to abide such treatment for their views. However, recent scholarship has also shown that there were places in Britain where anti-war views were relatively tolerated. Cyril Pearce's case study of Huddersfield justifiably concludes that the town was a 'resistor town'. This is too much of a complex argument to fully explain here, but basically COs in Huddersfield were tolerated, listened to, and sympathised with by a large amount of people. The town actually elected a well known CO as their mayor a few years after the war.

So to answer your question the most likely result of avoiding fighting in France would be extremely poor treatment from the authorities and most likely your own family, friends and the people of your community. However, perceptions did generally change to some degree towards the end of the war, and there were some notable areas where COs were well thought of and perceived as manly for resisting the authorities.

A couple of good sources:

Lois Bibbings, Telling Tales About Men: Conceptions of Conscientious Objectors to Military Service During the First World War (Manchester, 2011).

Cyril Pearce, Comrades in Conscience: The Story of an English Community's Opposition to the Great War (London, 2001; 2014).

There are also several journal articles by these scholars out there on similar topics and a fair few books and articles by different people on the Military Tribunals. The work of these two historians has revised a long held consensus that COs only received poor treatment and were only viewed negatively so any books you read published before 2000 are discussing strands of thought that have now been heavily revised.

[EDIT]: A couple of grammer/typo/spacing errors, and a new little bit at the end.