First, the name of the board game, or associated logos, is likely to be registered as a trademark in the UK. You can carry out a free search of the UK Trademarks Register. Use of a similar name may still amount to a trademark infringement.
Elements of a board game or logo are likely to be copyright or trademark protected as artistic work. Any counters on a board game may also be protected either under registered or unregistered design rights.
Finally, a similar game with the same overall look and feel of the original could trigger a ‘passing off’ claim (illegally implying there is a connection between you and another individual or company when no such link exists).
You should contact the manufacturers of the game or company in question to request their written permission to proceed rather than run the risk of being sued. You should explain your ideas and establish if they are willing to be supportive and / or grant you a royalty free licence, or grant this for a small fixed fee.
What legal issues should I be aware of, if I want to use a board game / company name or logo in my artwork?