You'll need to restrict the parameters so that the two curves meet at a single point:
t0uv = t0uv /. (Solve[A0uv - Auv Exp[-(t0uv/τ)] == B0uv + Buv Exp[-(t0uv/τ)],
t0uv] /. C[1] -> 0)[[1]]
(* τ Log[(Auv + Buv)/(A0uv - B0uv)] *)
For example, first generate some data:
SeedRandom[1235];
A0uvInit = 49.273;
AuvInit = 13.273;
τInit = 100;
B0uvInit = 37.006;
BuvInit = 36.472;
t0uvInit = 140;
data = Table[{t, Piecewise[{{A0uvInit - AuvInit Exp[-(t/τInit)], t < t0uvInit},
{B0uvInit + BuvInit Exp[-(t/τInit)], t >= t0uvInit}}]}, {t, 0, 250}];
data[[All, 2]] = data[[All, 2]] + RandomVariate[NormalDistribution[0, 1], 251];
Fit data to piecewise curve...
nlm = NonlinearModelFit[data,
Piecewise[{{A0uv - Auv Exp[-(t/τ)], t < t0uv},
{B0uv + Buv Exp[-(t/τ)], t >= t0uv}}],
{{A0uv, A0uvInit}, {Auv, AuvInit}, {B0uv, B0uvInit}, {Buv, BuvInit}, {τ, τInit}}, t];
nlm["BestFitParameters"]
(* {A0uv -> 49.59797043559751, Auv -> 13.465589303011921,
B0uv -> 36.869607720932656, Buv -> 34.24815518555274,
τ -> 103.17795183868176} *)
t0uv /. nlm["BestFitParameters"]
(* 136.33797461279067 *)
Plot the results
Show[ListPlot[data], Plot[nlm[t], {t, 0, 250}]]
t0uv = \[Tau] Log[(Auv + Buv)/(A0uv - B0uv)]
. This is from equating the two pieces att0uv
. $\endgroup$