I guess I will add to what I mentioned before. By going to the UK and logging in on your PC and performing work for a a US company and a being a non-resident of the UK and a US citizen, for the period of two months, you will not require a work visa in the UK. The key points are:
- You are not a Resident.
- You are not being paid by any entity in the UK
- You are not creating anything of value or performing a service for anyone in the UK
- You are not going to the UK for the sole purpose of working, your work does not require your presence in the UK.
All that said, as far as the UK is concerned...you are not working.
Even, as I mentioned before, if you were performing work duties...ie: Going to Meetings, giving presentations, doing research, actually a whole list of items, you still do not require a Work Visa. You can even perform paid work under certain circumstances, that is more limited, but still allowable. But do not take my word, please go here:
The key areas to look at are..."Permitted Activities for all visitors"
Even if you pick your purpose as "Business" and look for requirements, the key point is "Paid Work" meaning being paid to perform a service in the UK...you are not, and even then,m some paid work is allowed. However, as you stated, your purpose for going is not Business, you are just performing your normal US job responsibilities while visiting.