I would like to know how to restart a UK company which has been wound
up by company's house due to no tax returns being filed. This company
has not traded for 3 years and I would like to restart it to make use
of the capital gains allowance it has.

Contact Companies House (0870 3333636) who will advise.

Some years ago, I investigated the situation and I then got a positive
response but, inevitably, there were costs involved with the
probability that the Inland Revenue would require penalties.

I'm afraid it is near impossible to do this without the help of a
solicitor. There is some basic information on the Companies House
website. Click on FAQs, then "striking off, dissolution and
restoration". Then scroll down to the section which begins:

"How do I apply to restore a company to the Register?

A company may only be restored by a court order, so legal advice
should be sought."

Companies House
http://www.companieshouse.gov.uk/

Good luck!