How do we check if a company name is available or already has a trade mark?

Company names can be searched at Companies House.
UK trade marks can be searched on the UKIPO website.
EU trade marks can be searched on the EUIPO website. 

You should also do some unregistered searches to look for people who use a brand but don’t have it formally registered anywhere (so it’s not their company name nor a registered trade mark) as that can reveal names others are using which may lead to difficulty. All of that you can do yourself but if you want a more formal search, speak to an IP lawyer and they can arrange for searches (at a price) to be undertaken.