I have the Turnberry speakers and the Primaluna amps (although not together) and I think the speakers are best used with warmer sounding types of tube amplification. I am not a fan of Primaluna amps, but I've never heard the NAT amps so I cannot recommend then either. The NAT design looks interesting, and if they are available in your price range, they might work out.
For warmer sounding amps (something that would complement the Turnberry), I would look for amps run will 6L6 or KT66 tubes. I would tend to avoid KT88 amps because the tendency to have a "hard" or "brittle" sound might exaggerate the slight tendency toward sibilance of the Turnberry. Of course a 300B pushpull or single-ended amp will have the right tonal qualities, but, the issue would be sufficient power. I personally don't listen at very high volumes so my compromises would be in favor of low-powered alternatives, but, this is a personal decision.
These days, not that many amps are made with KT66 or 6L6 tubes because they have a lower output than other similar tubes making it harder to sell in a market driven by output specification. But, to me, they are the best sounding tubes of tetrode/pentode types. I also like EL84 amps, which are often the tube of choice in lower cost amps, but can still deliver lievly and punchy sound without being thin tonally.
For a specific choice, I would suggest looking at a Synthesis A40 amp (KT66). It not only sounds fantastic, it has a built in DAC that is no slouch.