Hi Sue. I've been down the being left on a minimal replacement dose of hormone road too - in my case after a total thyroidectomy when they sent me away for 3 months on 50mcg of T4. It's no fun. I've been 'lost' by the (irish) system on several occasions too.
All in all it tends not to pay to presume too much about their having it all under control.
It's also the case that high BP may not be directly linked to being low on thyroid - for example it can as in my own situation be the result of exaggerated stress responses/post traumatic stuff and the resulting high cortisol levels.
Too much thyroid can drive the BP up as well a too little, but in my case it's not a big effect (at least not at the very slightly excessive levels of hormone I've ever got to) - the BP has proven to be a pretty much separate phenomenon.
As ever a rational response requires getting down to spefics - to the bottom of what's causing your high blood pressure. There are of course causes that potentially would need care with the thyroid replacement.
The part I've found frustrating is that I've been through the hands of multiple doctors and consultant physicians (including several hospital admissions for high BP) - but not one of them got beyond fiddling with different combinations of BP pills and sending me home once boredom set in and my BP had dropped a bit. i.e. they never seemed to have a clue as to what was really causing it.
There are (if it proves to be what it looks like) fairly straight forward ways of treating my BP (which i've been posting about recently) - but this has emerged from personal research and naturopathic thinking, and not from mainstream medicine.
What I guess I'm suggesting is that it might be worth trying to do some digging to get under what's causing your blood pressure - in that it might clear the way for a rather more timely handling of your thyroid replacement.
Against that maybe you doc knows what the story is on your BP, and has good reason to go slowly on replacement....