If there are uncountable infinity of possible wavelengths of electromagnetic waves, does it imply an infinite bandwidth for information transfer?
More specifically and comprehensive. It's a complicated story, but nothing we've seen is infinite, BW is a pretty large resource but eventually depends on energy.
The bandwidth is the bandwidth (BW) of the signal that carries the information. There is indeed an infinite possible BW in nature, but you have to somehow have the signal generated. That takes power. You could theoretically (practically probably not, but let's indulge ourselves) have a power density of $\rho$ = exp(-kx) with X the freq, and when you integrate from X=0 to infinity you get 1/k. So you could do it, if you could design it that way, with no physical limit on BW. But information is not the same as BW, it is how many bits, or capacity, that channel could carry.
Shannon showed that, for classical information (ignore quantum for now, it comes up below) it is
C = BW * logbase2 (1+SNR).
SNR is $\rho$ * dW/($N_0$ * dW) for an infinitesimal bandwidth dW, with the other term the noise spectral density. The dWs cancel out and you have, after integrating $\rho$ over x SNR equal to 1/(k$N_0$). A finite constant. So, with finite power you could get an infinite C, for infinite BW.
So yes, BW is unlimited, and so is information capacity. But this is only classically, and only assuming you can generate an infinite BW of signals, or one with infinite BW.
The problems are physical and practical, and lead to quantum physics. The theoretical limit is quantum high energy physics. Creating a signal with infinite BW has not been done, or if it has, we are not able to detect the signal. Infinite BW requires a physical system that has energy produced over all freqs. Maybe they could theoretically, but at some point they are releasing one photon, and then just a random one occasionally. Not clear how that could be used. Alternatively, we need systems that can radiate over some limited BW, but at higher and higher, without limit, frequency. As you know, the quantum energy of a photon grows with freq, so we'd need to create signals with infinite energy, even if just one photon. That is, we really can't, practically. But we can get as high as the highest energy signals we can produce. Or if we want to be not so man-centric, the highest energy particles in the universe
For now, the highest energy we are able to produce is the 13 or so Tev's out of the LHC. But we are a long way from marshaling that to carry information. But even if we try to exploit natural processes at higher energies, there is still a limit.
Anyway, as you go to those higher energies where it is one particle (photon or something else) at a time, you need to use quantum information theory. At that point the best way to view it is as the quantum information in terms of qubits, and it can not be any bigger than some finite multiple of the number of particles. Down to the Planck limit, at this point in our know how of physics, the number of particles is finite in any finite region of the universe. So, only if the universe is infinite could information be infinite. We still don't know if the universe is infinite. We do know that the observable universe is finite. Thus, the observable information is finite.
Well, not so quick. We don't know what happens at the Planck scale, where quantum gravity reigns. Maybe the strings or whatever is the quantum at that scale has no energy limit. Well, we don't know.
Still, pretty clearly, there is a lot of BW that can be used, that we are not using.
People in the spectrum community, with theoretical preferences, are always saying the limits are how smart we are in using it, and in allocating the BW. People who have to create systems to send or store information are less optimistic, spend a lot of effort for any higher capacity they can get. And those always are fighting with the other laws of physics that keep information from propagating freely, and being detectable.