Your connections sound OK.
It may be that you have assigned/used the wrong input/output pins.
You say that you have had the servo working OK using the servo command so we can assume that side of things is OK.
That leaves the input.
Try this:-
do
pulsin 1,b0
debug
loop
You should see a value for b0 in the debug window which represents the pulse length comming from your radio.
If the value of b0 is zero, then your PICAXE is not seeing the pulses.
This could be for a number of reasons.
1. Wrong pin being tested.
2. Insufficient voltage change comming from radio.
If you have another PICAXE available, you could use that to supply servo pulses to your 28X1 input as a test.
What voltage is your radio running at?
What voltage is your PICAXE running at?
If the PICAXE supply is higher than the radio, it might not see the pulses.
Try running them from the same supply.
I don't suppose you have access to a 'scope?
If yes, have a look at the waveform comming from your radio.
What are the max and min voltage values?
The dalay issue:-
This realates to how your receiver decodes the signals, the order in which you read them and whatever processing you do.
Traditional RC equipment sends all the channel pulses serialy at frame rate of 20mS. It doesn't make any difference how fast your PICAXE runs, the frame rate remains the same. If you don't read the pulses in correct order, you can only hope to process alternate pulses at the absolute best.
Newer radios, in particular the latest 2.4Ghz systems send all channels at the same time. That means no matter how fast your PICAXE, you must wait for the next frame before the next channel can be read. Again, that results in a maximum of alternate frames for your mixer.
The very reason for the new method is to eliminate the 20mS delay.
A PICAXE mixer on the old system would at best run at a 20mS rate.
With the new system, it would run at best of a 40mS rate.
(slower if you mix more than 2 channels).
Hope you see the issue. Ask if it's still not clear.
If your mixer is for lights and/or something like undercarriage, then a small delay will not be a problem.
If it is for flight controls, then you WILL notice the delay.
As I'm sure you are aware, flying an RC heli is no simple task, don't make it harder!
For flight controls, it is better to do the mixing at the transmitter straight off the sticks.