Bayesian statistics is dumb. What's the point of using prior assumptions that are based on speculation? It's better to admitting your lack of knowledge, not jumping to conclusions without sufficient data, and processing data in an unbiased manner.
It is never proper to fabricate a biased (e.g. non-uniform) prior without any knowledge/information, that is not in any way part of Bayesian Inference. If you do have some extremely weak evidence, you use it accordingly with an extremely weak prior.
To paraphrase E.T. Jaynes, the rules of probability theory (e.g. Bayes Theorem) are the unique logically consistent way to reason about uncertainty.
Anyway, you can define a sequence of solutions with bounded uniform priors and calculate the limiting solution. For any given data set when the endpoints of the intervals go to +/-infinity the solution will converge to the uniform prior one - if it exist.
As someone that does Bayesian Inference a lot for my work (computational biology), I very often use uniform priors, but the structure of all real world problems I have ever encountered allows me specify hard bounds to the edges of non-zero probability.
A common example is when p(x|µ,v) is a Gaussian dist. with a prior on the mean set to p(µ)=1.
That isn't possible to use as a prior as the uniform distribution from negative to positive infinity is zero everywhere.
If you don’t think that this is possible that says more about you than about the shortcomings of Bayesian statistics.
As I said, you can define the improper uniform prior solution as the limit of a sequence of solutions corresponding to a sequence of increasingly wider intervals with endpoints that go to +/-infinity.
(And as I said, you can start with a suitably huge region. Say that you want to determine the position of something and use a uniform prior that extends to a distance of 10^27m - a perfectly bounded prior from a mathematical point of view that covers the whole observable universe. If you observe something outside it, it’s not with the prior that you have a problem.)
Failing that, I don't think there are any human usable data sources that could report observations over an infinite interval.