The NHS typically only test testosterone in women where high levels are suspected.
If Newson clinic want a blood level before treating you, which some providers do in order to make sure they aren't prescribing to someone who already has raised testosterone (however in reality this is as rare as rocking horse manure in the peri/meno population and frankly overkill in my opinion) then if you can't get it on the NHS somewhere like Medichecks or forthwith life will be able to help.
Super drug also do a testosterone service which may be cheaper.
I would also ask your GP to refer you to the local NHS menopause clinic for ongoing testosterone if they aren't willing to prescribe.
Antidepressants aren't a good treatment for hormone deficiencies, and can actually induce low libido and anorgasmia, so whilst you may have a bit of a hill to climb to get the treatment you want, there's no advantage to taking psychiatric medication unless you have a clinical indication for this.