Another vote from me on the cold tyres that have been sat on for a while. It should go once they get thoroughly warm though.
Other things to think of - slight play in wheel bearing(s)- easily sorted with a little nip up.
also if the tracking or caster is a little uneven you may get the wheels fighting each other. If the front tie bar bushes are worn you can easily get toe out once moving at speed and this will give some instability for the rest of the set up to deal with.
also check lower steering column UJ as first signs of this going is that little bit of play in the steering.

If it is tyres, can you swap the wheels over from side to side? or turn 180deg on each hub? see if it makes a difference.
Rob