Anything over 100ml IS a problem at LHR; they do confiscate contact lens solution larger than 100ml in carry on luggage as it does not fall into the medical exemption category there.
I don't see where OP is flying from the US.
From what I understand, OP is flying SIN-LHR, taking a train, and possibly LHR-SIN again.
If he does decide to take it in carry on, then the rules in SIN and LHR apply, and the rule at LHR is no contact lens fluid if larger than 100ml in hand luggage (ie carry on)