The TM symbol is used when an application for trademark is made with the trademark registry. The TM symbol is thus used to indicate the fact that a trademark application exists with respect to the trademark and serves as a warning for infringers and counter-fitters.
Registering a domain name and company name through Companies House is crucial. However, these do not prevent anyone else from using your business name. A registered trade mark will prevent anyone else from using your company name. It will also prevent anyone else from registering a trademark for your business name.
The Poor Man's Patent Is Obsolete Being the first to invent will no longer save you is someone else filed first. So even if you did write out the idea for your invention and mailed it to yourself, that date would not matter.