What do you plan to do with the estimated price? Just some reporting, or do you intend to execute at this price?
(For a few USD bonds, you may be able to find people making markets in London and Asian hours, but not very likely.)
Instead of using treasury for risk-free rate, I'd use the swap curve. For non-callable bonds, just calculate Z-spread using the last available bond price and the USD swap curve at that time. Some hours later, assume for starters that the Z-spread has not changed and back out the price from the Z-spread the London morning USD swap curve. Recalculate when the swap curve moves. For callable bonds you could use OAS... I don't see any obvious problems using that.
You could improve on the assumption that the Z-spread doesn't change, for exampe, if you observe how this bond's Z-spread changes with CDX.NA.IG or .HY spreads, and if you see the index London morning, assume that your Z-spread change was consistent with its history. I doubt that you can get single-name CDS spreads or do anything useful with them.
You can backtest your predicted prices by comparing with the prices that actualy appea NY morning.
Please note that it would be a very bad idea to post these prices on some e-trading venue. If some news comes out that changes the Z-spread by a lot, then people will see that your Z-spread hasn't moved and will hit/lift your prices.