1485 Tri Club is affiliated to Cycling Time Trials - http://www.ctt.org.uk/
This means that as members of the club you can enter any Open or Club Time Trials. Just give the Tri club name when entering / signing on. Most popular distances are 10 and 25 miles. Riders are set off at 1 minute intervals. No drafting is permitted.
Open events typically take place on Saturday afternoons (10 miles) and Sunday mornings (25 miles +). Details of events can be found on the Cycling Time Trials site. Events have to be entered 10 days in advance. You have to fill out a form that you download from the site and send it with a cheque of around £8.
Of more interest for training purposes are Club events. These take place on week day evenings from April through till September. You just need to turn up on the night at the signing on point, pay your £2 or £3, pin on a number and ride.
Hinckley Cycle Racing Club organise a 10 mile Time Trial every Tuesday night from the Axe and Compass pub near Wolvey. These start on Tuesday 7th April with the first few weeks starting at 18:30 (sign-on from 18:00 for HCRC members and 18:15 for guests - this is us) before switching to 19:00. Be warned that you must ride with your shoulders covered (they are less worried about heads). Some of us are members of HCRC and will be down there and can explain the details.
If you get bored of riding up and down the same bit of road or cannot make Tuesdays, there are other club events listed on http://www.midlanddc.org.uk/. The other clubs also organise some 25s. I have found all of the Coventry and Rugby clubs very friendly when riding as a guest.
Another option is the Leicester Forest club event on the A46 on a Thursday evening http://www.leicesterforest.org.uk/.