I'd suggest your first step should be to lock down your budget. If you want to spend £200 or under, then your two quality options are going to be between two of the three main 7 inch tablets - the Google Nexus 7 or the Kindle Fire HD (the iPad Mini being the 3rd option, but it is over £200).
To get a quality 10 inch tablet, you are going to need a budget of around £400. Yes, there are no-brand 10 inch tablets available on Ebay for under £200, but I think you'll really struggle to find a friend or family member (or anyone!) who has bought one and is happy to vouch for it. Google, Amazon and Apple tablets are like the John Lewis version of a tablet - they will be good quality and reliable, and if you're not happy with them you can take them back and get a refund. You can find research reviews of Google, Amazon and Apple tablets from all the mainstream newspapers and other media, unlike the no-brand Ebay tablets which definitely go under the "buyer beware" category.
If you are looking to go with a Nexus 7, and one of prime uses for it will be loading it full of tv and movies for the kids, I'd really suggest you look at spending £200 on the 32GB version to give you space.
play.google.com/store/devices/details?id=nexus_7_16gb&feature=microsite
Buying an 8GB version for £150 is a false economy as the films and tv shows will fit it up too quickly, and you'll be constantly deleting things and re-loading them to meet your kids' demands for variety.
If you're not technical and don't plan investing time to fully understand how to do stuff on the device, do think about stretching to £350 for the 32GB iPad Mini. The extra £150 gets you the simplest device to use, plus the backup of being able to stroll into any Apple shop and ask for help if you get stuck.
Apple charge a premium for their stuff, and you can look at it as either a "sucker tax" for people bewitched by the brand, or an investment in after sale support knowing that they'll stand behind anything they sell you. Again, it's very similar to the John Lewis approach to retailing.
But if there is a hard limit on your budget - focus on the Nexus 7 and the Kindle Fire HD.