If it's all in writing you can't just force another company to do what you want. What you can do is wriggle, twist and delay until it becomes too expensive for the smaller company to continue to pursue.

However judges are more than well aware of this technique and will allow the plaintiff to accrue costs against Legal Aid (paid for by government).

So what usually happens:

1. Small co files against large co for using same name
2. Large co produces huge response document which is all piss and wind
3. Small co says they can't afford the costs to answer each point
4. Judge permits Small Co to use Legal Aid.
5. Large co offers to settle. (E.g. you're a 3 person sandwich shop. They offer you £10m. No more work, no more hassle)

If Small Co is energetic, young and courageous, they may choose to fight to the death. But Legal Aid has a limit...