You could order the microtainer version of the Thyroid Plus 10. This involves pricking your fingers (with lancets that are supplied) and dripping the blood into little vials (also supplied). Instructions are provided. The vials get packaged up, taken to the Post Office on Monday to Thursday (not Friday - Sunday), and postage is paid for "guaranteed next day delivery before 1pm".
You could order the vacutainer version of the Thyroid Plus 10, and for this you are supplied with test tubes (bigger than the microtainer ones), and everything else you need to get blood taken from the crook of the elbow. Then the tubes have to be packaged and posted as above. To get blood taken for you there are a few choices, but they all cost money unless you have a friend who can do it for you.
I have used my local NHS hospital phlebotomy department to get blood taken. Phone up in advance and ask if they will do it for you. Expect it to cost money, and ask how much it is. It varies quite a lot from hospital to hospital. Once the blood is taken package it up as soon as possible and take it to the Post Office. There may be a cut-off time beyond which you can't get guaranteed next day delivery, so find out in advance when that is, and allow time for queueing.
Get the blood samples done as early as possible in the day, ideally before 9am. Fast overnight and before the blood is taken, except for water which you can drink freely.
If you normally take thyroid meds, then leave 24 hours between your last dose (assuming it is levo) and taking the blood. The time to leave is slightly different for T3 or NDT, but I can never remember what the rules are for those.