They could have built some but those would have had relatively little impact on the blockade, although they would have made attacks on US ports more difficult. Even then however it would be a case of them being located in the ports their built in, which would have to occur before those ports were successfully attacked as they were very much brown water boats in the most part. As such Britain could concentrate a force to overwhelm any single point while the US forces would be find it a lot more difficult to link up.One can't know how a war will play out. The US could build a fleet of very heavily armored Monitor class craft that could cause havoc to the RN.
We do know that when Lincoln put his foot down on the British sale of Laird Ram' s to
the Confederacy the UK government did back down.