Water companies are (from 1st October last year) responsible for more sewer pipes than thy used to be. It's possibly that it might be their responsibility to fix the problem (at their expense), rather than yours. Click on the relevant type of property, and then on the date of construction, here:
https://www.anglianwater.co.uk/household/wastewater-services/Private-sewers-and-lateral-drains.aspx
That will show you which pipes you're responsible for. (It doesn't matter that you're probably not with Anglia Water. The same rules apply across the country).
If it's your responsibility to fix the problem, your cheapest option is probably to hire a set of drain rods from a company like HSS. Prices here:
http://www.hss.com/g/51410/Drain-Rods-And-Keys.html
Instructions here:
http://www.hss.com/how-to-unblock-a-drain
Branch locator here:
http://www.hss.com/tpd/multimapstorefinder/Branch_Finder.html
Failing that, AVOID using the 'big boys' (such as DynoRod) unless you've just won the EuroMillions jackpot. Look in your local parish magazine, community newsletter or similar for a qualified plumber, who should be able to do the job for under a hundred quid (unless it's particularly tricky and, for example, he has to hire specialist jetting equipment).
Chris