The electrical items must carry 'CE' labels, to confirm that they meet the relevant European standards. See Section 9 of the legislation
http://www.legislation.gov.uk/uksi/1994/3260/made
Self-adhesive 'CE' labels can be purchased on eBay.
As long as the construction is generally sound, a simply Portable Appliance Test should suffice to check that your items are safe. Find a charity shop that sells electrical items and ask them who does their PAT tests. If it's (say) a local electrician who does them on a voluntary basis he might well do your tests for you in return for a small donation to the charity. (Otherwise google 'PAT test', together with the name of your area, to find someone who can do the job for you).
The law relating to plugs on electrical devices can be found here:
http://www.legislation.gov.uk/uksi/1994/1768/made
While mass-produced items are now fitted with moulded plugs, I can foresee no problems if you simply used the replacement screw-fit ones that are sold in all hardware stores, as long as the plug
(a) conforms to the current legal standards (as any plug purchased from, say, B&Q automatically will anyway) ; and
(b) is correctly fused for the load of the appliance (which, for the items you've referred to, means 3A, not 13A).
With regard to shades, there's no requirement for any kind of formal testing, as it's the manufacturer (i.e. you) who declares them to be safe. They should carry the CE mark and a label to indicate the maximum wattage of a bulb that they can safely be used with. (With modern bulbs producing far less heat than old-fashioned tungsten bulbs, that might seem to be less important than it was. However you should assume that the purchaser of the shade might still be using hot tungsten bulbs).