This comes from an AstraZeneca press release, but it sounds like there's strong evidence that the AZ vaccine doesn't cause blood clots and this is a typical case of "if anything bad happens after you take a drug, it gets reported as a side-effect", which is why one of the possible adverse effects of the Moderna vaccine is getting struck by lightning:

A careful review of all available safety data of more than 17 million people vaccinated in the European Union (EU) and UK with COVID-19 Vaccine AstraZeneca has shown no evidence of an increased risk of pulmonary embolism, deep vein thrombosis (DVT) or thrombocytopenia, in any defined age group, gender, batch or in any particular country.

So far across the EU and UK, there have been 15 events of DVT and 22 events of pulmonary embolism reported among those given the vaccine, based on the number of cases the Company has received as of 8 March. This is much lower than would be expected to occur naturally in a general population of this size and is similar across other licensed COVID-19 vaccines.

(Emphasis added)

From Twitter, it looks like the rates of clotting-related issues in UK recipients of the AstraZeneca and Pfizer/BioNTech vaccines are essentially indistinguishable.

If there's an actual risk of blood clot problems, we should see something similar reported in the UK, given the millions of doses they already used.

We don't see any similar reports coming from the UK. The opposite seems true (BBC).

Is the UK hiding this kind of news? I doubt.

Are there problem only outside of the UK? 
If this is the case, "the story" might actually be some handling/logistic problem in the vaccines production/transportation/storage in other countries that turns some batches bad.

If this is the case we should see higher % of problems where the batches are mishandled.
I could not find numbers about this, it looks like at the first problem some countries are halting the distribution of entire batches.

A clearer smoking gun would be multiple people having blood clots problems after receiving doses from the same batch.
The size of the batch would help estimate how unlikely that event is.

Italy halted ABV5811 and ABV2856, but unfortunately I could not find how big these batches actually are. If you have any idea I'd be glad to know.

I'd feel completely safe to get a vaccine in the UK.
In lack of the specific batch information, I'd feel safe outside of the UK too.

Off the top of my head, it could easily be the case that we still need to rule out that the groups getting the vaccine are already more prone to blood clots. In particular, I seem to remember some talk about Covid infections leading to blood clots, and many of the early vaccinations have been given to folks most likely to be exposed to Covid. We've also seen that "asymptomatic" (referring specifically to the symptoms of cough, fever, & anosmia) Covid seems to be really common in the group of all infections.

It's not much of a stretch to guess that -- if the blood clot problem turns out to be real -- it may be at least partly because of those "asymptomatic" infections among the groups we've targeted for early vaccination specifically because they are more likely to be exposed (and therefore more likely to have already been infected without knowing) than other groups.

There's a Twitter thread claiming that there is statistical evidence that blood clots are unusually common in Germany after getting the AstraZeneca vaccine:

The PEI ran an "observed-versus-expected analysis", comparing the # of such cases expected without vaccination in a 14-day period with the # of cases reported in those vaccinated with #AstraZeneca (1.6 million) in Germany. About 1 case would be expected, 7 were observed

I still think stopping vaccination is insane for several reasons:

• According to this source, approximate 200 people are dying of COVID per day in Germany. Even if we assume that 6 out of every 1.6 million people who get the AZ vaccine get blood clots and die, that would be 365 deaths total in Germany or 2 days worth of COVID deaths.

• According to this source, the survival rate of cerebral venous-sinus thrombosis is ~95%, and taking that into account, even if the 6 / 1.6 million number is correct, we would expect only 1/20th of those people to die of it, leading to approximately 20 people in Germany dying of this (again, vs ~200 people per day dying of COVID).

• I'm still very skeptical that this 6 / 1.6 million number is actually correct since it's a very small number and small numbers are possible to get by chance, especially if you're doing multiple comparisons: Why are we only looking at the number for Germany? Wouldn't this be reported if it had happened in France, or the UK, or...?

A couple of days ago I came across this deposition to the "German Corona Extra-Parliamentary Inquiry Committee", by Dr. Vanessa Schmidt-Kruger, who describes herself as "a cell biologist and my specialist field is the functional characterisation and elucidation of proteins".

She enumerates a range of concerns with the BioNTech vaccine, including purity; toxicity of lipid nanoparticles; dosing and long-term immune effects. There is an almost throw-away comment in there to the effect of: "...it has also been found that when the LNPs are transported in the blood then thromboses can occur, or haemolysis – haemolysis means the sudden dissolution of erythrocytes, i.e. red blood cells, this causes hypoxia."

I'm not in a position to evaluate any of this -- the deposition is quite long, and technical (altho note that the second half of the doc is just the German version). The presentation.. somewhat twinges my "this might be crackpot" aesethetic sense (as per naming-the-nameless), but I can't dismiss it out of hand based on that.