Edinburgh to Bath via London

Read this page. Looks like if the ticket is marked with cross (symbol looks like a Catholic cross), the tube transfer is included with the ticket.
http://www.nationalrail.co.uk/times_fares/ticket_types/46587.aspx

The tube times are just for guidance - you're not expected to get those particular tube trains as they are so frequent, just get the first that arrives.

You will have enough time.

You are making it more complicated by travelling via London - which involve a tube joinery from Kings Cross to Paddington. Direct trains (www.crosscountrytrains.co.uk) go from Edinburgh to Bristol (Temple Meads) - which is only a short 15 minute hop by another train to Bath. Alternatively, you could use Virgin Trains from Edinburgh to Birmingham and then change for a train to Bristol.
https://www.virgintrains.co.uk

If using the former route mentioned above (or going via London), I would consider an overnight at York - assuming you are not doing that going north.
Try out the options at www.nationalrail.co.uk (Fares rise nearer date of travel - pre book specific trains about 11 weeks ahead if you can and avoid Sundays (engineering work & Friday’s -- very busy so fares tend to be higher). It may be cheaper to split the tickets. Try splits at York / Derby / Cheltenham and Birmingham if using Virgin.

It may pay you to get (on arrival) a Family Railcard in order to reduce the rail fares. (You can pre book saying that you already have the Railcard). http://www.familyandfriends-railcard.co.uk

Also consider flying from Edinburgh to Bristol - which has bus link to Bath. www.easyjet.com

It is often cheaper to travel via London, mainly because there is a relative lack of cheap fares on Crosscountry, and what there are tend to be snapped up very quickly.