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I was surprised and grateful to get 14 interesting replies to this
obscure topic and have summarised them in case there are relevant to
other celiacs with thyroid disease.

The general view confirmed that becoming hyperthyroid after being
hypothyroid was very unusual.

There was a consensus that the known links between the two conditions
could be affecting the progress of my thyroid disease but opinion was
split between whether this might be positive (by removing gluten) or
negative (continued malabsorption). Most thought that my gluten-free
diet could be helping my thyroid to "recover" but some thought that
absorption problems, particularly a 'leaky gut', could still be
affecting my thyroid adversely and causing the hyperthyroidism.
However, in my case I should point out that I was GF well before the
thyroid problem started (strictly GF after celiac diagnosis for over 2
years, biopsy-confirmed diet-compliant and EMA negative for 1 year), and
that a year ago my thyroid was near normal (TSH 2.9).  The diagnosis
remains Hashimoto's thyroiditis. I can believe that my GFdiet could have
possibly helped the thyroid to recover over the summer but the puzzle is
that I am now hyperthyroid (on no thyroxine) and this is certainly not
expected or normal even for celiacs.

However it is still early days and as I am otherwise very healthy I hope
and expect an improvement.  I shall report any developments.
Many thanks again to all those who passed on information and
speculation.

SUMMARY: (my comments in brackets)

1 A number of people confirmed the connection between thyroid
abnormalities and celiac disease and the belief that a gluten-free diet
influences the progress of autoimmune thyroiditis.  Among them Ron
Hoggan who has already posted his references which are, as ever,
interesting and relevant.

2 A gluten-free diet has reduced the need for thyroid medications for
several celiacs, in one case quite dramatically from 3mg Synthroid to
.125mg after being GF for 20 months.  8 Conversely 2 people thought
absorption could still be a problem if newly diagnosed, leading to
hyperthyroidism (but I was GF before any thyroid problems)

3 Several people suggested I have Hashimoto's thyroiditis (which is
correct) and that this is hard to control. One person had fluctuating
thyroid levels for 30 years!

4 Someone else knew of a case of similar swings but said that eventually
the hyperthyroid periods exhaust the thyroid leading to permanent
hypothyroidism. (But I thought this only happened when the initial
diagnosis was the autoimmune hyperthyroidism known as Graves' Disease ?)

5 Thyroid medications should always be introduced slowly, 25mg at a time
(this could be relevant to me as I went on to 100mg quite quickly)

6 Fluctuating thyroid problems in women can to linked to diet and
menopause, as explained by Dr Gillespie from the useful
thyroidabout.com. website( I am 47 but my hormones have been checked
out. However I have looked at her diet pages before and found them
interesting though not totally I am not totally convinced of the medical
evidence for the problems with soya)

7 One retired MD had come across a similar case of Hashimoto's to mine
30 years ago, when the autoimmune connections were still unknown.

8 Sometimes a viral illness can produce temporary hypothroidism. (It is
not clear if this includes Hashimoto's but it would fit some of the
strange viral-like symptoms, including swollen lymph glands, that I had
at the time of diagnosis. This is an interesting angle but does not
explain my currently over-active thyroid)

9 Dr Fasano might be of use (unfortunately being in the UK this is not
an option) or see an endocrinologist if problems continue (which I hope
to do but the GP's referral could take some months)

Charlotte Ward-Perkins, Oxford UK