These are a few simple ideas, following up on Jamie's comment about the expected rapidity distribution p_L for produced W's (where p_L refers to longitudinal momentum, not polarization).  Besides some kinematic smearing due to partonic motion within the protons, similar to the p_T one might expect for the W's created, there is also a real p_L distribution due to the differing x's of the interacting partons.  For W production at the Tevatron, one might expect the fusing q and qbar to have similar Bjorken-x distributions; but in a pp collision, these would usually be very asymmetric.  Some simple kinematics can help our intuition (at least for me), so I just calculated some relevant quantities for a given value of x_quark:

In Jan's toy model, it was assumed that the W was created (on average) at rest along the z axis. Physically, this would be the case when the quark and anti-quark each carry in 40 GeV/c of momentum, or x₁ = x₂ = 0.16 for a 250 GeV proton beam. A more typical event might involve a quark with x₁ = 0.32 hitting an anti-quark of x₂ = 0.08, which yields a W with 60 GeV/c of longitudinal momentum, and a total energy of 100 GeV. Events of this sort would yield electrons (in the lab frame) of varying total energy depending on theta (or eta), see below. Their p_T distribution would be skewed in angle, but still peak right at 40 GeV.  As an example: for a lepton emitted at 90 deg in the W rest frame,we can make the plot below.

We can also compare the electron energy and transverse momentum as a function of lab angle for beta=0 (W at rest, x₁ = 0.16) and the case discussed above for beta = 0.60 (W has 60 GeV momentum, x₁ = 0.32):

For W⁺ production, one could take current PDF's for u(x) and dbar(x) and fold them together to get a better sense of what kinematics we are actually sampling. Presumably we can use RHICBOS to do all of this for us.