Should I change my domain name if there is a company with the same name?

I suppose you haven't launched your busines yet. To avoid future legal battle, here is my suggestion:

Register a new domain name, register your company with the new name. Start your business using this new name.

BUT, keep the ownership of the conflicting name for a year. Just in case that company wants to buy your domain name, you can ask for a price.

Hard to know without knowing additional information - like how generic or specific the name you are talking about is. If it were me, I'd probably launch forth with my preferred name. I'm not a lawyer, but I believe in the US there is no law against operating a name with the same name as another company as long as you aren't exploiting the other company's brand assets in some way. Presumably it is similar in the UK.

In this case, it sounds like there are no assets of theirs there for you to exploit (and vice versa). The other company is apparently not much further along than you and certainly doesn't exist on the domain name you just bought.

You might check some other TLDs to see if they might be operating under another extension. You might want to also check a trademarks database, google, and so forth to be double sure they aren't somehow actively using the name.

In my opinion, you are both at about the same point (they have a registered company, you have a domain name) and whoever makes the name known first is the winner (assuming there even needs to be a winner - possibly you can both coexist with the same name in different spheres).

Get a new domain name (cheap) to avoid the potential (expensive) legal hassle down the line.

Good names that aren't taken are hard to come by and desired domain names are hard to find too. What about using a slightly modified name for a company name? So if your domain name is "AwesomeBrand.com" your company name can be "Awesome Brand [something]" (e.g. " & Co." or "International"). It seems it's not all that rare to use only a part of the company name for a domain name since length is important.

C'mon community, lets come up with some suggestions for extending the name! This seems like it could be a common enough problem where the answer can become evergreen and help many.