To provide "evidence" of using the name from a certain time and going forward, you can simply file a fictitious name statement at your local county recorder's office. Fees vary by county. I've found the folks there very helpful for a couple of businesses I've done this for. This is the procedure for what's known as "DBA," or "doing business as," which you need complete to enable ppl write checks to the business name anyway.
THEN, sometime down the line, you can pay the bux to do trademark. For that, I think the search fees -- to see that another like biz entity doesn't have the name you want to use -- are what'll get ya (statewide, nationwide, international ... it adds up).
...... Damn, I just noticed you're in the UK. Don't know what the laws are there and if there's a similar "fictitious name" procedure, but you might want to look into that.