Can only speak from personal experience which is limited but there is most certainly differences in SD quality amongst different full hd tvs so its a fair assumption that this would extend to full hd vs hd ready. Ultimately it depends on the actual tv and whats under the hood and to an extent your perception of adequate PQ. I'm sure there are some hd ready sets out there that do a better job than full hd ones in the SD department but haven't seen any first hand.
If I understand correctly Sky HD broadcasts in 1080i so if you have a HD ready tv there will always be an element of downscaling involved and depending on your tv the results may vary. From reading the forums these past 9 months I have seen so many different opinions on output settings for their tv and Sky HD that I put a lot of things down to personal preference and perception.
Intrestingly I have seen an older Panasonic LCD produce better PQ with SD Sky & Scart than Full HD sets displaying SD content via Sky HD using HDMI. So I am sure there are a lot of people who are disappointed with SD content when they upgrade to Sky HD.
Without knowing what tv you own its hard to tell if you will reap any benefits from a new tv if you are concerned about quality of SD. You may be getting great SD PQ without realising it and you expectations are to high
Comparing the two full hd sets i've owned, LG42LH3000 & Sony 40W5500 the difference is remarkable when watching SD content on Sky HD via HDMI. The LG made most SD channels appear 'blocky', abnormalities in the picture were frequent and magnified to the extent it put me off watching SD content completely.
Thanks to a two dodgy LG's I upgraded to the Sony and was shocked to discover the difference in SD PQ, especially as I only paid an extra £30 for the Sony which is in a different league.
Unless you have money to burn you may have to make some compromises when choosing the right tv for you. I was limited to a budget of £550 which made me go for the LG as this was the best I could find within budget (at the time the Sony was close a grand!). You also have to really think about the main function of the tv and what content you view and take any planned purchases (blu-ray, games consols etc) into consideration. I didn't have Sky HD when I bought the LG and had no intention to do so, I was living in a cable area with "free cable" the PQ with my PS3 was mind blowing and intended this to be the main source for the tv mostly blu-ray and downloaded hd content. Due to the nature of my cable source I just settled for having a picture at all and didnt worry about quality, so there were no probs as far as I was concerned. Then I moved to a non cable area so I took out Sky HD and it was then I started finding fault with the LG especially with Sky SD content.
Now my main use is Sky HD content (Sports then Movies) then Sky SD channels, Blu-Ray (PS3) and gaming (PS3). The W5500 is notorious for lagging issues with PS3 but a compramise I was willing to make when picking a different tv to the LG as the results with Sky HD (HD & SD channels) are fantastic. So if gaming was a big thing for you you may have to be prepared to make a compromise in the SD PQ department and vice versa. I would be surprised if there weren't any tvs where some compromise would have to be made if your budget extends to bottom/mid range tvs and you were using multiple sources. If you solely use SKY HD then it may be easier to find the 'perfect' tv for you.
It a bit unfair to compare the LG & Sony but it does show that you can get get tvs that produce great results with SKY HD SD content for a bargain price. At the end of the day it does all depends on your budget. The best thing to do would be find a decent retailer who allows you to see multiple sources and SD content on their tvs, most of the bigger retailers I've ventured into will insist on only showing you blu ray and say that they are unable to hook their tvs up to Sky probably in an attempt to hide the poor pq when handling sd.
Oh and move your armchair back, 2 feet!!!