what's the baseband FFT size of LTE?
Read the book 4G Lte Advanced by Stefan Parkvall and Eric Dahlman. According to the book,
FFT size of 2048, with a corresponding sampling rate of 30.72 MHz, is suitable for the wider LTE carrier bandwidths, such as bandwidths of the order of 15 MHz and above. However, for smaller carrier bandwidths, a smaller FFT size and a correspondingly lower sampling rate can very well be used.(Lowering the sample rate simplifies the implementation in dual-mode HSPA/LTE terminals)

Also,nothing forbids the implementation of an OFDM modulator as a set of parallel modulators, 
Also, nothing prevents the use of a larger IFFT size, for example a size-2048 IFFT size, even in the case of a smaller number of OFDM subcarriers.

Tcp= 160·Ts and Tcpe=512.Ts. Change in CP induces change in the total number of REs(6X12, instead of 7X12). Note that #sub carriers remain 12 in both cases and FFT size is dependent on BW allocated which is expressed in #sub carriers.