If you want F1 work experience, you aren't going to get it without contacts, and you certainly aren't going to get in now - only two teams actually have an application process for work experience - Williams and McLaren. McLarens deadline for work experience this summer passed last year, and Williams passed at the very start of the year so you are waaaaay too late to apply to any team now. GP2, Formula Two and FR3.5 are huge no-no's too, again waay too late. You need to be looking way lower, so F3, FR2.0 and GP3 at a stretch might consider you. Your best bet is to look at the junior categories that race on the weekends of events like BTCC and F2. Also look for companies outside of motorsport teams, but that work in motorsport. There are thousands of companies, you just have to work to find them! Google is the best place to start really, look through series websites, go to the teams websites, send CVs/covering letters off to the teams, then look at the companies they work with and send CVs off to them too.
Also, whilst I'm at it, if you want to go into F1, you're an idiot if you don't pick anywhere other than Hertfordshire. We have more graduates in F1 than any other university in the world. Plus theres an F2 driver on the course. Plus we're the only university in the world to have an actual F1 simulator (the same as three F1 teams). Plus F1 teams actually come to our university asking for graduates to employ!! It's mega!!!